Steady State Vibrations of a Piecewise Linear System
Bibliographic record
Abstract
Abstract In this paper we examine the dynamics of a mechanical system with a piecewise linear damping and stiffness, which is characterized as a strongly nonlinear system. Searching for the exact steady state periodic response of piecewise linear systems for harmonic excitations, lead to a set of transcendental equations which should be solved numerically. The strong nonlinear characteristic of the system is the reason for failure of most traditional perturbation methods in providing an approximated frequency response. In this paper we show that the averaging method of Krylov-Bogoliobuv-Mitropolsky (or multiple time scale method) may be used to find the behavior of this type of system. Optimization of linear vibration isolators and engine mounts has a straightforward solution: the stiffness should be as soft as possible. Static deflection and maximum allowable relative mount displacement (rattle space) determine the limit of mount softness. Dividing the travel space into two (or more) stages enables us to use an isolator with a different stiffness for each stage. This is a simple method to have a more advanced passive mount isolator. However, this also makes the system piecewise linear, which has a strongly nonlinear behavior if the ratio of the two stages are far from unity. Asymmetrical bilinear systems, with different stiffness and damping values for compression and tension, are used when the relative motion in negative direction is more (or less) critical than in positive direction. Sensitivity analysis of the system shows that the difference in the damping plays a more important role than the stiffness difference in these systems. In this paper, using averaging method, we examine the nonlinear behavior of piecewise linear mounts at resonance, and provide a comprehensive analysis which may be used to better design for these kind of vibration isolators.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.004 | 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".