Using a piecewise linear spring to approximate an essentially nonlinear spring: design and validation
Bibliographic record
Abstract
This study develops a procedure for designing a piecewise linear spring (PLS) to approximate an essentially nonlinear spring (ENS). The PLS is constructed with a cantilever beam constrained by a pair of single- or double-stop blocks. The design begins by determining the restoring force of the desired ENS using the equivalent stiffness which quantifies the characteristics of a cubic polynomial. Then, based on the force-deflection model of a cantilever beam with an overhang, the configuration parameters for single- and double-stop blocks are determined through a least squares optimization. The numerical simulation demonstrates that the PLS with double-stop blocks approximates the desired ENS behaviors better in terms of the restoring force, potential energy, and instantaneous frequency transition. An experiment apparatus with four tunable stop blocks is developed to validate the numerical simulation results. The static experimental tests verify the accuracy of the analytical model. The dynamic experimental tests show that within the achievable range of displacement, the PLSs behave similarly to the ENS. However, the maximum displacement is smaller than the designed one due to an insufficient exciting force. To address this issue, the desired displacement range is reduced by half. With the redesigned PLSs, the improved experimental results are obtained.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.001 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".