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Gear root bending strength: statistical treatment of Single Tooth Bending Fatigue tests results

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Zahnfußbiegefestigkeit: Statistische Behandlung der Ergebnisse von den STBF-Versuchen

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Abstract

Gear tooth failure due to tooth root bending fatigue is one of the most dangerous failure modes in gears. Therefore, the precise definition of gear bending fatigue strength is a key aspect. Furthermore, in order to calculate the service life of a gear component, an accurate estimation of the S‑N curve is required. This curve must properly account for the slope of the fatigue strength region, the slope of the region ahead the fatigue knee, as well as the position of the knee itself. Hence, in order obtain a curve at different reliability levels, a proper estimation of the dispersion associated with the experimental points is required. Usually, Single Tooth Bending Fatigue (STBF) tests are used to investigate the gear load carrying capacity with respect to the bending failure mode. Then, starting from the gear specimen test data, the S‑N curve that has to be used in the rating method has to be determined. Maximum Likelihood Estimation (MLE), a statistical tool capable of considering also interrupted tests (e.g. runouts) has been used to estimate, in the most reliable way, the S‑N curve from experimental points. Furthermore, if the STBF test is performed in a symmetrical configuration, i.e. two teeth loaded at the same time, also the survival of one of the two teeth represents an information that can be taken into account thanks to MLE.

Moreover, additional effects must be considered to obtain the S‑N curve of a real component starting from STBF tests. In reality, the load history and the statistical behaviour are different, since in a meshing gear the strength is determined by its weakest tooth, while in a STBF test the failing tooth is predetermined. The latter effect is considered by means of the statistic of extremes, which enables the estimation of the strength of the weakest tooth and therefore of the gear. This paper describes in detail the proposed calculation method and explains its application to determine the S‑N curve in practical cases.

Zusammenfassung

Versagen von Verzahnungen aufgrund von Zahnfußbiegeermüdung ist eine der gefährlichsten Versagensarten von Zahnrädern. Daher ist die genaue Definition der Biegewechselfestigkeit von Getrieben ein wichtiger Aspekt. Zur Berechnung der Lebensdauer einer Komponente eines Getriebes ist die genaue Schätzung der Wöhlerkurve erforderlich. Diese Kurve muss die Neigung des Ermüdungsfestigkeitsbereichs, die Neigung des Bereichs der Ermüdungsknie sowie die Position des Knies selbst richtig berücksichtigen. Um eine Kurve mit unterschiedlichen Zuverlässigkeitsniveaus zu erhalten, ist daher eine zuverlässige Schätzung der Streuung in Verbindung mit den experimentellen Punkten erforderlich. Üblicherweise werden Single-Tooth Bending-Fatigue(STBF)-Versuche verwendet, um die Getriebetragfähigkeit in Bezug auf den Biegeversagensmodus zu untersuchen. Ausgehend von den Prüfdaten des Zahnradmusters muss dann die Wöhlerkurve bestimmt werden, die im Bewertungsverfahren verwendet werden muss. Die Maximum-Likelihood-Schätzung (MLE), ein statistisches Werkzeug, das auch unterbrochene Tests verwaltet: wird der STBF-Versuch in einer symmetrischen Konfiguration durchgeführt, d. h. zwei Zähne gleichzeitig belastet, stellt auch das Überleben eines der beiden Zähne eine Information dar, die dank der MLE berücksichtigt werden kann.

Darüber hinaus müssen zusätzliche Effekte berücksichtigt werden, um ausgehend von STBF-Versuche die Wöhlerlinie eines realen Bauteils zu erhalten. In Wirklichkeit sind die Belastungshistorie und das statistische Verhalten unterschiedlich, da bei einem Zahnrad die Festigkeit durch den schwächsten Zahn bestimmt wird, während bei einem STBF-Versuch der ausfallende Zahn vorgegeben wird. Letzterer Effekt wird durch die Extremwertstatistik berücksichtigt, die eine Abschätzung der Festigkeit des schwächsten Zahnes und damit des Zahnrades ermöglicht. Dieses Veröffentlichung beschreibt ausführlich das vorgeschlagene Berechnungsverfahren und erläutert seine Anwendung zur Bestimmung der Wöhlerlinie in praktischen Fällen.

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Acknowledgements

The authors express their gratitude prof. Stefano Beretta for his invaluable support during the validation of this statistical model.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Politecnico di Milano, Milan, Italy

    Luca Bonaiti, Francesco Rosa, Prasad Mahendra Rao & Carlo Gorla

  2. Faculty of Science and Technology, Free University of Bolzano, Bolzano, Italy

    Franco Concli

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  1. Luca Bonaiti
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  2. Francesco Rosa
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  3. Prasad Mahendra Rao
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  4. Franco Concli
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  5. Carlo Gorla
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Correspondence to Luca Bonaiti.

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Conflict of interest

L. Bonaiti, F. Rosa, P. M. Rao, F. Concli and C. Gorla declare that they have no competing interests.

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Bonaiti, L., Rosa, F., Rao, P.M. et al. Gear root bending strength: statistical treatment of Single Tooth Bending Fatigue tests results. Forsch Ingenieurwes 86, 251–258 (2022). https://doi.org/10.1007/s10010-021-00567-7

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