Closed-form performance-efficiency (PE) formulas for three- and four-stage differential gear transmissions whose speed is controlled through the sun gears were derived by Strilets, Malashchenko and Strilets (2019) using the potential-power method, with every gear stage represented by a fixed transmission efficiency (numerically η13(i)(4)=0.97\eta_{13(i)}^{(4)}=0.97) and with identical stage parameters in all published numerical examples, and were checked only against the authors' own MS Excel computations. That study is purely theoretical and computational and lists no remaining work; three open directions follow from the documented limitations of its model and define the transition from this theoretical assessment to a validated, design-ready energy-effectiveness model: (1) specification and justification of an experimental validation and benchmarking program for the analytical/numerical PE predictions under real loading, bearing and lubrication conditions; (2) replacement of the fixed per-stage efficiency parameter by loss contributions that depend on tooth geometry, contact ratios, bearing losses and lubricant effects; and (3) extension of the analytic loss model to higher stage counts and to non-identical per-stage parameter distributions (non-identical ratios u13(i)(4)u_{13(i)}^{(4)} and unequal control speeds), including assessment of the design limits at which the simplifying assumptions of the present derivations fail.

Background

Many machine drives need continuous control of the output speed. Stepped gearboxes achieve it discretely and generate dynamic loads at each shift, while classical stepless (continuously variable) transmissions rely on friction connections that wear and transmit limited torque. To avoid both drawbacks, a Ukrainian research group (Malashchenko, O. Strilets, V. Strilets and co-workers) has, since the mid-2000s, developed speed-change devices based on a gear differential — a planetary (epicyclic) gear train consisting of a central sun gear, planet gears, an internal ring gear, and a carrier holding the planets. In these devices one link of the differential is connected to a closed-circuit hydraulic system (a gear pump feeding a loop with a control valve/throttle) that acts as a controllable rotary stopper: throttling the loop changes the speed of that control link and hence the output speed continuously, without friction clutches. The control link can be the carrier, the ring gear, or the sun gear of each stage; coupling several single-stage differentials in series gives a multistage transmission with a wider control range. In the sun-gear-controlled variant studied in the source paper, the first-stage carrier (or ring gear) is the driving link, the last-stage ring gear (or carrier) is the driven link, and the sun gears 1(i)1_{(i)} of the individual stages are the control links.

The energy effectiveness of such a transmission is measured by its overall performance efficiency (PE), the ratio of output power to input power, and a central design question is whether the device can enter self-braking — a regime in which the transmission absorbs input power without delivering useful output (PE at or below zero). The group computes PE analytically by the 'potential power' method, a classical technique for planetary trains in which the friction losses of each gear mesh are taken to be proportional to the power transmitted in the motion relative to the carrier; each stage is therefore represented by a single fixed transmission efficiency parameter, η13(i)(4)\eta_{13(i)}^{(4)} (numerically set to 0.97 in all published examples), independent of load, speed, tooth geometry and lubrication. In this notation the subscript 13 denotes the sun-gear 1 / ring-gear 3 mesh of the stage, the superscript (4) denotes the converted mechanism with the carrier 4 stopped — that is, the fixed-carrier gear ratio u13(i)(4)u_{13(i)}^{(4)} and the corresponding fixed-carrier efficiency η13(i)(4)\eta_{13(i)}^{(4)} of stage (i)(i) — and the index (i)(i) numbers the stages; the superscript is therefore not a stage count, and the same symbols are used for three- and for four-stage transmissions. Single- and two-stage efficiency formulas for this device family were derived in earlier papers by the same group (2016–2017), and a companion paper (Diagnostyka 2019) treated the carrier-controlled variant for two and three stages; in its review of prior work that paper notes, of a third-party bi-planetary transmission design, that the geometry and strength of gears, shafts and rolling bearings were omitted from the calculation as typical design work, and it concludes of its own results that they are a basis for further power calculations and experimental research of gear differential transmissions of speed-change devices with carrier control.

The source paper (Naukovyi Visnyk Natsionalnoho Hirnychoho Universytetu, 2019, issue 6) reports, for the first time, closed-form PE formulas for three- and four-stage sun-gear-controlled differential transmissions with a driving carrier and driven ring gear, or vice versa, together with numerical maps produced in MS Excel: driving-link speeds of 75–300 rad/s, sun-gear control speeds of 0–50 rad/s (taken equal across stages), per-stage ratios u13(i)(4)u_{13(i)}^{(4)} from 1 to 10, and η13(i)(4)=0.97\eta_{13(i)}^{(4)}=0.97. Its stated tasks are to obtain analytical expressions for PE determination, to get graphic dependences of PE on the angular velocity of the sun gear, the ratio and the number of stages, and to draw conclusions about the energy effectiveness of such transmissions and the possibilities of self-braking; its conclusions are that overall PE decreases as the number of stages grows, that self-braking is not possible in any of the examined cases because the PE is far greater than zero, and that the obtained PE values can be used at the design stage, with implementation in design and construction practice and use in the educational process given as the practical value. The paper further states that its method can evaluate the energy effectiveness of higher step number differential gear transmissions — an extension it does not carry out. Two features of the methodology limit its predictive power for real hardware: every stage's loss is a constant parameter unrelated to tooth geometry, contact ratio, bearing design or lubrication; and no comparison with any physical measurement exists — the only verification is the authors' own numerical evaluation of their formulas. A third restriction concerns the published results rather than the derivation: the closed-form expressions keep distinct per-stage symbols throughout, but every numerical example is computed for identical stages (same ratio, same efficiency, equal control speeds), so non-identical per-stage parameter distributions have never been evaluated. Related methodology that could lift these restrictions exists in the general planetary-gear literature (for example, experimental efficiency analysis of two-degree-of-freedom epicyclic transmissions by Chen and Chen 2015, and detailed tooth-geometry-, bearing- and lubricant-dependent power-loss models for planetary gear sets by Nutakor et al. 2017), but it has never been integrated into the analytical model of this device family, nor has any prototype measurement of the sun-gear-controlled multistage differentials been published.

Problem Statement

Advance the energy-effectiveness assessment of multistage sun-gear-controlled differential speed-change devices (driving first-stage carrier or ring gear, driven last-stage ring gear or carrier, sun gears as control links) from the current purely analytical, fixed-parameter state to a validated, design-ready predictive model. The source work (Strilets, Malashchenko & Strilets 2019) completed a theoretical and computer study, recommends its formulas for design and teaching use, and lists no remaining work; the three scientifically linked open directions below are therefore not tasks stated in that paper, but follow from the limitations of its model documented above and together constitute this objective:

(1) Specification and justification of an experimental validation and benchmarking program for the analytical/numerical PE (performance efficiency) predictions of the three- and four-stage configurations under real loading, bearing and lubrication conditions (measurands, instrumentation, loading and lubrication conditions), together with quantification of the expected sensitivity of predicted PE to those conditions; the derived conclusions that PE decreases as the number of stages increases and that no self-braking occurs in the examined operating ranges (ω1(i)=0\omega_{1(i)}=0–50 rad/s control speeds, drive speeds 75–300 rad/s, per-stage ratios u13(i)(4)u_{13(i)}^{(4)} of 1–10) are the claims the program is to test.

(2) Replacement of the fixed per-stage efficiency parameter η13(i)(4)=0.97\eta_{13(i)}^{(4)}=0.97 by loss contributions that depend on detailed tooth geometry, contact ratios, bearing losses and lubricant effects, integrated into the analytical model of the multistage sun-gear-controlled differential so that predicted PE reflects the actual design and operating conditions.

(3) Extension of the analytic loss model to higher numbers of stages (beyond four) — the source asserts that its method covers higher step numbers but does not perform the extension — and evaluation of non-identical per-stage parameter distributions (non-identical ratios u13(i)(4)u_{13(i)}^{(4)} and unequal sun-gear control speeds), which the closed forms already admit symbolically but which have never been computed, together with an assessment of the design limits at which kinematic couplings or component constraints invalidate the simplifying assumptions used in the existing derivations (fixed per-stage efficiency, identical stage parameters in the published results).

Directions (1)–(3) are mutually dependent, and none of them alone resolves the objective: (a) the experimental validation program of direction (1), specified and justified together with the claims it is to test and the sensitivity quantification; (b) the geometry-, contact-ratio-, bearing- and lubricant-dependent loss contributions of direction (2) integrated into the analytical PE model, so that predicted PE reflects the actual design and operating conditions; and (c) closed-form or numerically evaluated PE expressions for configurations beyond four stages and for non-identical per-stage parameter sets, together with the corresponding self-braking assessment and the design-limit assessment of direction (3).

References

  1. Strilets, O. R., Malashchenko, V. O., & Strilets, V. M. (2019). Energy effectiveness of the differential of a device for speed change through the sun gear. Naukovyi Visnyk Natsionalnoho Hirnychoho Universytetu, 2019(6). DOI: 10.29202/nvngu/2019-6/10. URL: https://doi.org/10.29202/nvngu/2019-6/10 (journal page: http://nvngu.in.ua/index.php/en/authors-and-readers/journal-headlines/1830-engcat/archive/2019/contens-6-2019/5079-energy-effectiveness-of-the-differential-of-a-device-for-speed-change-through-the-sun-gear)
  2. Malashchenko, V., Strilets, O., Strilets, V., & Kłysz, S. (2019). Investigation of the energy effectiveness of multistage differential gears when the speed is changed by the carrier. Diagnostyka, 20(4), 57–64. DOI: 10.29354/diag/112397. URL: https://doi.org/10.29354/diag/112397
  3. Malashchenko, V., Strilets, O., & Strilets, V. (2018). Theoretical foundations for the development of experimental equipment for determining the load capacity of speed change devices. Ukrainian Journal of Mechanical Engineering and Materials Science, 4(1), 125–130. DOI: 10.23939/ujmems2018.01.125. URL: https://doi.org/10.23939/ujmems2018.01.125
  4. Strilets, O., Malashchenko, V., & Kłysz, S. (2021). Review and complement of methods for changing the movement speed of mechanisms and machines elements. Ukrainian Journal of Mechanical Engineering and Materials Science, 7(1–2), 9–22. DOI: 10.23939/ujmems2021.01-02.009. URL: https://doi.org/10.23939/ujmems2021.01-02.009
  5. Strilets, O., Malashchenko, V., & Strilets, V. (2020). Dynamic model of a closed-loop hydraulic system for speed control through gear differential. Scientific Journal of the Ternopil National Technical University, 98(2), 91–98. DOI: 10.33108/visnyk_tntu2020.02.091. URL: https://doi.org/10.33108/visnyk_tntu2020.02.091
  6. Strilets, O., Malashchenko, V., & Pasika, V. (2021). Algorithm for designing of speed change control devices through a gear differential with a closed-loop hydraulic system. Scientific Journal of the Ternopil National Technical University, 1(101), 138–148. DOI: 10.33108/visnyk_tntu2021.01.138. URL: https://doi.org/10.33108/visnyk_tntu2021.01.138
  7. Malashchenko, V. O., Strilets, O. R., Strilets, V. M., Lutsyk, V. L., Smolarz, A., Droździel, P., Torgesizova, A., & Shortanbayeva, A. (2021). Energy efficiency of gear differentials in devices for speed change control through a carrier. In Mechatronic Systems 1 (pp. 255–265). CRC Press. DOI: 10.1201/9781003224136-22. URL: https://doi.org/10.1201/9781003224136-22
  8. Malashchenko, V., Strilets, O., & Strilets, V. (2016). Method and device for speed change by the epicyclic gear train with stepped-planet gear set. Research Works of the Air Force Institute of Technology, 38(1), 13–20. DOI: 10.1515/afit-2016-0002. URL: https://doi.org/10.1515/afit-2016-0002
  9. Malashchenko, V., Strilets, O., & Strilets, V. (2017). Determining performance efficiency of the differential in a device for speed change through epicycle [sic — published title; the article analyzes an epicyclic gear train]. Eastern-European Journal of Enterprise Technologies, 6(7(90)), 51–57. DOI: 10.15587/1729-4061.2017.110683. URL: https://doi.org/10.15587/1729-4061.2017.110683
  10. Nutakor, C., Kłodowski, A., Sopanen, J., Mikkola, A., & Pedrero, J. I. (2017). Planetary gear sets power loss modeling: Application to wind turbines. Tribology International, 105, 42–54. DOI: 10.1016/j.triboint.2016.09.029. URL: https://doi.org/10.1016/j.triboint.2016.09.029
  11. Chen, C., & Chen, J. (2015). Efficiency analysis of two degrees of freedom epicyclic gear transmission and experimental validation. Mechanism and Machine Theory, 87, 115–130. DOI: 10.1016/j.mechmachtheory.2014.12.017. URL: https://doi.org/10.1016/j.mechmachtheory.2014.12.017
  12. Strilets, O., Malashchenko, O. [sic — Volodymyr O. Malashchenko; given name is misprinted in the publisher metadata], & Stepaniuk, A. (2021). Application of three-dimensional modeling to optimize the dimensions [sic — 'DIMENTIONS' as published] of the speed control device with rotary stopper in the form of a closed-loop hydraulic system. Bulletin of the National Technical University «KhPI» Series: Engineering and CAD, 1, 91–96. DOI: 10.20998/2079-0775.2021.1.11. URL: https://doi.org/10.20998/2079-0775.2021.1.11