Algebraic Differential Kinematics of Planar 4R Linkages
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Bibliographic record
Abstract
In this paper some new and some already established results are discussed which are ideally suited for teaching four-bar mechanism kinematics to senior undergraduate mechanical engineering students. We re-examine the velocity and acceleration level kinematics of planar 4R linkages for the six distinct angle pairings of the four link orientation angles, θ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> -θ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</inf> , in the light of the first two time derivatives of the resulting algebraic equations. Freudenstein's Theorem 1, which states that the angular velocity ratio of the two ground-fixed moving links is equivalent to a ratio of the three instantaneous centres of velocity aligned on the line joining the centres of the two ground-fixed R-pairs, is re-expressed in terms of the coupler line equation via the algebraic input-output (IO) equation, and confirmed using the first time derivative of the same equation. We prove that similar ratios can be easily obtained for the five other distinct IO equations for any planar 4R linkage. A clear advantage of our novel approach is that we obtain six signed ratios, thereby indicating the relative sense of the angular velocities, whereas the generalised Freudenstein theorem 1 reveals only four. We next identify conditions for minimum and maximum angular velocities and propose a corollary to Freudenstein's Theorem 2. Finally, we investigate the acceleration level kinematics for all six IO equations, determine extreme values, and discuss implications for extreme values of torque and force transfer as well as shaking forces.
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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 | 0.000 | 0.000 |
| 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 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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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