Tooth Mesh Research Articles

Analysis of electromechanical coupling vibration characteristics of motor-gear system based on bond graph theory

In recent years, there has been a growing demand in the field of engineering systems for research on the vibration response of motor-gear systems under non-steady state conditions. However, conventional modeling and simulation techniques are primarily restricted to systems within a single energy domain. When dealing with intricate electromechanical coupling systems, the modeling process becomes increasingly intricate. Consequently, this study adopts bond graph theory as the foundation for analyzing the vibration characteristics of motor-gear coupling systems. In addressing the vibration issue of the motor-gear transmission system, a nonlinear bond graph model is established that takes into account the time-varying mesh stiffness, static errors, and tooth surface friction, among other nonlinear factors. Then, according to the coupling relationship between motor mechanical systems and electrical systems, the electromechanical coupling dynamic model of the motor-gear system is established using the method of bond graph. In addition, the state equation of the system is derived based on the bond graph model. Finally, the system is simulated using 20-sim software, and the electromechanical coupled vibration characteristics of the simulation results are analyzed. This article lays the foundation for the design and analysis of motor-gear coupling systems and multi-energy domain complex systems.

Calculation of the deformation values of the ammortizer in the compound angle gear transmission

The article presents the method of calculating the axial deformations of the rubber bush used in the bevel gear with the recommended composition and the formulas for determining the axial deformations. Based on the analysis of the connection graphs, the recommended values of its parameters were determined. The article presents the results of an analysis of the design features and disadvantages of cylindrical gears. An effective design scheme for a helical cylindrical gear transmission with elastic elements has been developed, allowing it to absorb peak values of oscillations of torques and axial force components arising from the meshing of wheel teeth, as well as cycling noise indicators.

Theoretical modeling and transmission characteristics analysis of a novel double-roller hourglass worm drive based on enveloping principle

Traditional planar double-enveloping hourglass worm pairs have the characteristics such as the strong bearing capacity, high transmission efficiency and long life-span. However, the high relative sliding velocity and the serious agglutination or abrasion may also occur at the contact point on tooth surfaces. In this paper, a novel type of double-roller hourglass worm pair is proposed and analyzed, wherein, the worm part adopts from the planar double-enveloping worm pair, the corresponding meshing tooth surface of worm gear is replaced by two rows of cylinder rollers for reducing friction and enhancing meshing efficiency. That is, the original meshing tooth surfaces in worm gear are replaced by the common tangent plane of rollers to form a special type of meshing drive, the original line-contact form between tooth surfaces was changed to point-contact so that the sliding friction was transformed into rolling friction, and the meshing efficiency can be greatly improved. The theoretical model of double-roller hourglass worm pair was established based on spatial enveloping principle, the meshing equation and tooth surface equations were derived. The solid models of worm and gear were established and assembled. The parameters of enveloping surface angle, roller radius and friction coefficient, which affecting relative motion velocity at meshing contact point and efficiency during the whole transmission process was analyzed and discussed systematically. The results showed that comparing to the traditional worm pair, the roller’s rotation greatly reduced the relative sliding velocity and improved meshing efficiency at the contact point in this double-roller hourglass worm pair.

Influence of ring gear flexibility on the fatigue reliability of planetary gear systems in heavy helicopters

Flexibility of ring gear rims significantly affects the mechanical properties of planetary transmissions at both levels of gear tooth meshing and system power sharing, and is one of the important design factors determining the service reliability of large aviation planetary systems. Therefore, to analyze the mechanism by which the flexibility of ring gear rims affects the fatigue reliability of planetary systems, in the framework of the stress-strength interference theory, a reliability assessment model for the planetary system is constructed based on a calculation logic extension for the full probability formula. Starting from balancing evaluation accuracy and evaluation cost, a hybrid finite element method and a gear tooth lifetime transformation test are employed to provide load and strength input variables for the reliability model respectively. Finally, a mapping relationship from ring gear rim flexibility to planetary system fatigue reliability is established. For the specific design scheme of planetary systems, a lower limit of the ring gear rim thickness can be determined according to the requirements of reliability indexes, so that the balance between reliability and lightweight can be achieved to some extent.

Adult patient expectations and satisfaction: Can they be influenced by viewing the three-dimensional predicted outcome before fixed orthodontic treatment of dental crowding?

Investigating the possible changes in patients' expectations of and satisfaction with the orthodontic treatment outcomes when they were given the three-dimensional digital prediction of their teeth alignment before the beginning of treatment. A prospective non-controlled single-group clinical trial was conducted on 28 (18 females, 10 males, mean age: 20.68 ± 1.91 years) patients with Class I malocclusion and moderate dental crowding who required a nonextraction orthodontic treatment. Patients were given the expectations questionnaire on their first visit (T0). Then, patients were shown a three-dimensional digital setup-created by Orthoanalyzer software (3Shape, Copenhagen, Denmark)-of the proposed treatment results before orthodontic treatment (T1) and received two questionnaires, the second expectations questionnaire and the satisfaction with the proposed changes questionnaire. Treatment was then initiated using the fixed appliances and completed. After debonding, a question about patients' satisfaction with the achieved changes was completed (T2). Patients' expectations level significantly increased after watching the predicted alignment of teeth compared with the initial levels in terms of chewing (x¯=5.54 and x¯= 6.71), speech (x‾ =5.93 and x¯= 6.93), and oral hygiene improvement (x‾= 7.93 and x¯=8.61 ± 1.06; at T0 and T1, respectively). The remaining items showed no significant differences between the two assessment times. Patients had a higher level of satisfaction at the end of treatment than after watching the proposed outcome. There were significant statistical differences in the items related to teeth appearance (x¯= 9.12 and x¯= 9.60 at T1 and T2, respectively), and teeth engagement (x¯= 8.92 and x¯= 9.40 at T1 and T2, respectively) CONCLUSIONS: Watching the predicted alignment outcome increased patients' expectations of chewing, speech, and oral hygiene improvement compared with the initially recorded levels. Patients were more satisfied with the final result than what was recorded after viewing the predicted plan. However, these results were not clinically significant.

Impact of Manufacturing Errors on Tooth Surface Forces in RV Reducer Cycloid Pinwheels

The mechanical calculation model of rotary-vertical (RV) reducer is established on the basis of considering displacement and stiffness. Without considering the manufacturing error, the parameters of the RV reducer are input into the model and solved by MATLAB calculation software. Comparing the solution results with the analysis results based on Hertz contact theory, the error is within a reasonable range, which verifies the reliability of the mechanical calculation model. The manufacturing errors of the RV reducer based on the high-precision measuring instrument were imported into the mechanical calculation model for solving, the results of the force analysis of the tooth surface of the cycloidal pinwheel considering the manufacturing errors were obtained. The maximum value of the load was 1713N, and the number of meshing teeth was 17 pairs. Compared with the case without manufacturing error, the maximum value of load increased by 89% and the number of meshed teeth decreased by two pairs. Based on the existing experimental data, the force on the tooth surface of the cycloid pinwheel after considering the manufacturing error is closer to the actual situation.

Experimental investigation of moving contact pattern in helical planetary gearboxes

Planetary gear sets combine the characteristics of high-power density and compact design which are beneficial for a wide range of high-performance applications. Embedded in the efforts for resource efficiency and cost reduction, new challenges arise regarding a lightweight design. An accurate prediction of the load sharing on the individual planetary gears and especially of the load distribution of the meshing gears is of major importance. A uniform load distribution enables the design of the individual tooth meshes without load increases and oversizing. However, a varying load distribution has been observed in planetary gearboxes with respect to the rotation of the carrier which is caused by misaligned central shafts of the gearbox. This behavior is influenced by several factors, such as manufacturing and assembly deviations, the rigidity of shafts, housing and bearings, the number of planets, the quality of the gear wheels and the operating conditions. To determine the exact load distribution, detailed simulations or extensive experimental prototype tests are required. In an early design phase, these detailed boundary conditions are not yet defined and their effect on a moving contact pattern therefore cannot be reliably considered.Thus, in this paper an experimental investigation of the influencing factors on a varying load distribution with regard to the carrier rotation is presented. Based on experimental data recommendations for the design of planetary gearboxes are derived.

Acoustical behavior of periodic flank modifications under dynamic operating conditions

Noise emissions belong to the crucial design characteristics in modern transmissions. The most efficient way to achieve beneficial noise behavior is to reduce the excitation in the tooth contact, since the tooth contact represents the main source of noise in gearboxes. Special flank modifications may optimize the noise behavior, but the common design approach uses static assumptions, whereas operating conditions include dynamic operation.The application of periodic flank modifications as a special type of flank form provides a possibility to avoid the typical compromise between the flank design objectives load carrying capacity and low excitation behavior which results from the exclusive usage of standard modifications such as crownings or tip- and root relieves. The principle of periodic modifications is based on the direct compensation of the inconstant part of the elastic deformations in the tooth mesh. For this reason, the static loaded transmission error (LTE) is typically used as specific value to evaluate the effect of these flank forms on the excitation. In theoretical and experimental studies, the effectiveness of periodic modifications to optimize the excitation behavior could be verified. However, the analysis of these modifications under dynamic operating conditions is still scarcely documented. In this paper, the mesh excitation of periodic and standard modifications under various dynamic operating conditions in the subcritical, resonance and supercritical operating range is simulated and compared to each other. In addition, experimental investigations at the dynamic test rig of the Gear Research Center (FZG) were performed. Due to the measurement of the torsional acceleration level during continuous speed runups, the simulated results can be validated by experimental data.

Dynamic characteristics of the rack and pinion lift mechanism of the combined command cabin

This paper presents a novel approach to accurately simulate the dynamic characteristics of a rack and pinion lifting mechanism while accounting for the coupling effect of meshing tooth pair stiffness and grease characteristics in real-world environments. A structure-grease coupling meshing stiffness model is established based on the deformation coordination conditions of the tooth pair meshing. The model incorporates the effect of the grease stiffness term on the time-varying meshing stiffness of the rack and pinion. By analyzing the dynamic characteristics of the combined command cabin lifting mechanism using this model, we find that the total meshing stiffness of the gear teeth is lower when the transient thermoelastic flow effect of the grease is considered. Furthermore, the total stiffness value decreases as the normal meshing force at the meshing point decreases. This research reveals that the worst lubrication conditions are typically found on the gear wheel teeth near the base circle, where the grease film temperature rise is highest, the grease film pressure is highest, and the grease film thickness is thinnest.

Impact of phase configuration of a helical idler gear system on the vibration

The helical idler gear system has a lot of applications for reducer. The number of idler teeth and the orientation angle between the center lines of the three gears directly determine the mesh phasing. However, their impact is often ignored in system design. This paper presents the phase configuration relationship of the idler gear system, which can be used to control the mesh phasing by adjusting the number of idler teeth and orientation angle, to achieve vibration reduction. By establishing the dynamic model of an idler gear system considering phase configuration, the effects of the idler teeth number, orientation angle, and mesh phasing on the dynamic characteristics are investigated. For odd and even idler teeth numbers, the change law of mesh phasing with orientation angle is revealed. When the orientation angle is constant, the number of idler gear teeth will directly affect the system vibration response from the mesh phasing. The mesh phasing and its vibration response change periodically with the orientation angle. In each cycle, there is an optimal mesh phasing that reduces the variation response of different modes, and its value of different periods basically remains the same. In a period of mesh phasing, the suppression mechanism of mesh phasing under different vibration modes is studied. The variation law of the optimal mesh phasing with the rotational speed is revealed, and the vibration reduction effect can reach 20% on average. Finally, the correctness of vibration response and mesh phasing relationship is verified by experiments and a rigid-flexible coupling model. The above research provides a foundation for realizing vibration reduction by adjusting phase configuration.

Dynamic modeling of spur gear system under marine ship heaving-pitching motion

Sea waves often result in multi-freedom motions of ships, and affect the stability of marine gear systems. This study presents a novel dynamic modeling method for spur gear system considering the influence of marine ship heaving-pitching motion and investigates the spur gear dynamics. An excitation transmission path from wave to gear is illustrated and incorporated into the gear dynamic model. The hydrodynamic force is analyzed and the hull heaving-pitching motion is solved using the New Strip Method (NSM). Considering the time-varying meshing stiffness, transmission error, and tooth side clearance, the motion differential equation of the spur gear system impacted by the heaving-pitching coupling motion of the hull is established. The influences of key wave and navigation parameters on gear dynamics are analyzed. The results show that ship heaving-pitching motion causes additional stiffness and damping excitation, and leads to typical nonlinear phenomena of the spur gear system in a marine ship, such as period, quasi-period, and chaos, and causes a sharp increase in the meshing impact amplitude up to a maximum of 36.81%. The changes in wave frequency, the hull's heading angle, and hull's speed have a significant impact on gear dynamics, whereas the wave amplitude has less of an impact.

A double-layer iterative analytical model for mesh stiffness and load distribution of early-stage cracked gear based on the slicing method

To reveal the coupling relationship between the time-varying mesh stiffness (TVMS) and the load distribution along the tooth width direction (TWD) of gears with early-stage crack (ESC), a double-layer iterative analytical model for the TVMS and load distributions of gears is proposed considering the effects of the non-uniformly distributed load (NDL) along TWD caused by the ESC. In the proposed model, an analytical model of tooth torsional deformation and a parallel slice stiffness model of the tooth pair with ESC are separately developed based on the slicing method. On this basis, a double-layer iterative calculation method for the TVMS and load distributions is proposed, in which the coupling relationships between the slice stiffness and load distribution along TWD as well as the TVMS and load distribution between the meshing tooth pairs are respectively presented with the inner- and outer- layer iterations. Finite element (FE) models are established to verify the proposed double-layer iterative model. The effects of crack parameters and applied torque on the tooth torsional deformation, TVMS, and load distributions of the gear with ESC are finally investigated based on the proposed model. The results show that the proposed model can realize the accurate and fast decoupling calculation of the TVMS and load distributions of gears with ESC. This study can provide a basis for the establishment of the refined ESC fault diagnosis method and the rapid evaluation of the load distributions of gears with asymmetric errors or faults along TWD.

Research on time-varying meshing stiffness of wind turbine gearbox considering tooth surface wear

The time-varying meshing stiffness (TVMS) is an important element of the gear system. In this paper, the TVMS of each gear set of a wind turbine gearbox is calculated based on the principle of potential energy method, where wear is the most likely failure in the operation of a wind turbine gearbox. Therefore, the Archard wear model is brought into it, and theoretically the analytical formula of stiffness considering tooth wear is derived, and the time-varying wear depth is deduced as the natural wind speed keeps changing and the input torque of the wind turbine gearbox keeps changing. The interrelationship between gear tooth wear and meshing pressure angle and number of meshes is studied in depth, and the TVMS of each gear set under wear is calculated using the modified tooth profile model. The results show that the tooth profile wear depth is related to the gear meshing angle, and the tooth profile wear depth becomes larger as the number of meshes increases, thus reducing the TVMS of the wind turbine, and the TVMS shows irregular fluctuations under time-varying wear.

Contact stress analysis of external non-variable multi-modulus spur gear pairs

The conditions of correct gear engagement of a pair of involute gears and the principle of geometric relationship of involute tooth profile are derived to determine the pitch point of externally meshing non-variable multi-modulus spur gear pairs into the double tooth engagement region. The Hertzian contact stress model is used as the basis to derive the contact stress calculation equations for pitch point and other characteristic contact points, and the variation of the contact stress distribution on the contact line with the module ratio is analyzed. The results demonstrate that when the number of teeth of the gear pairs is determined, the larger the value of the module difference, the easier it is for the pitch point of the gear pairs to be in the double tooth meshing zone. Except for the double tooth meshing out point, the contact stress decreases with the increase of module ratio, and the maximum point of contact stress is at the upper boundary point of the single tooth meshing zone.

Определение участка нарезания мелкомодульных зубьев в эксцентриковых механизмах свободного хода

Normal forces use to transfer the load makes it possible to increase load capacity and reliability of the free running mechanisms. Normal forces in the eccentric free running mechanisms are realized in engagement of the fine-module teeth that are cut on the outer cage inner surface and the outer surface of the eccentric rings. To ensure correct operation of the mechanism in jamming and wedging, value of the fine-module teeth cutting section angle should be selected from the equality condition of the radial clearance between the outer cage teeth and the eccentric ring in the entire working section. To solve the problem, calculation scheme is proposed and mathematical model is obtained that describes relationship between the fine-module cutting section angle and the mechanism geometric parameters. The nature of the main geometric parameters influence (radius, eccentricity, radial clearance, module, etc.) on the value of the teeth cutting section angle was established. It is shown that eccentricity provides the greatest influence on the working area size, radial clearance and module influences are less pronounced.

An improved load distribution model for involute helical gear pairs considering an actual contact ratio

With the increasing requirements of helical gears for high strength and low noise, it is significant to improve the accuracy of the load distribution ratio depending on the minimum elastic potential energy (MEPE) model. In this paper, an improved distribution model is developed to replace the theoretical contact ratio with the actual contact ratio and introduce the potential energy of the fillet foundation for comparison. Then, the operation process of the helical gear is divided into seven cases. Finally, combined with the finite element method, the accuracy of the improved model is confirmed. Results indicate that the load distribution model with fillet foundation energy is only effective under the front or the full tooth meshing. When the gear meshing includes the rear part of a tooth, the inclusion of the actual contact ratio and fillet foundation energy should be considered.

Numerical Analysis of Friction Reduction and ATSLB Capacity of Lubricated MTS with Textured Micro-Elements

The simulation analysis numerically investigates the thermoelastic lubricated interfacial Textured Micro-Element (TME) load-bearing contact, a theoretical model is proposed, and the effective friction reduction and Anti-Thermoelastic Scuffing Load bearing (ATSLB) capacity between random rough Meshing Teeth Surfaces (MTS) are presented, the mechanism linking interfacial thermoelastic lubrication, TME meshing friction reduction and ATSLB is revealed. The real contact domain area between MTS with multi-scale Micro-Element Textures (MET) is obtained for the numerical calculation of the three-dimensional equivalent TME contact volume, which is the correlation bridge between friction reduction and ATSLB of the thermoelastic lubrication interface. The proposed theoretical model predicts the time-varying behaviour of the textured meshing interface friction reduction with TME contact load under thermoelastic lubrication conditions. Numerical simulations show that the textured interface meshing volume is the key to solving the load-bearing problem of line contact between randomly rough teeth surfaces. The friction coefficients of the MTS are reduced by 13–24%. The lubricated load-bearing and friction reduction behaviour between the textured MTS is quantified by the thermoelastic voids of TME interface and actual meshing volume ratio, which provides a new perspective for further insight into the lubrication and friction reduction behaviour between the MTS with multi-scale MET-ATSLB coupling mechanism.

Non-Linear Dynamic Modeling and Analysis of Split-Torque Face-Gear Drive Systems

Abstract A non-linear dynamic model of a multi-mesh involute spur pinion-driven face-gear split-torque drive system is developed. The lumped mass model consists of five pinions and two face-gears with seven rotational degrees-of-freedom. Model includes two inputs, two outputs, and three idler gears meshing with two identical face-gears at the bottom and top of the assembly. Face-gear tooth form and corresponding spur gear teeth are established by differential geometry and the theory of gearing. Face-gear drive system mesh stiffnesses are established by utilizing finite strip method for the first time in the literature. The mesh stiffness and damping are time-variant. The model includes clearance-type non-linearity. The harmonic balance method is utilized to solve the non-linear differential equations of motion. The results are compared with the time simulation results. The stability is checked with Floquet theory and bifurcation diagrams from Poincare sections. The model is fully capable of generating the tooth geometries, mesh stiffnesses, and gear train dynamics without reliance on any package programs. Hence, the model provides flexibility and fast computation for parametric studies compared to existing literature. The effect of the pinions’ orientations on the face-gear is investigated to evaluate the mesh phasing effects among system dynamic response. The effect of sub-harmonic motion on the dynamic response of the split-torque face-gear system is also demonstrated for the first time in the literature. The case studies demonstrate that sub-harmonic resonance peaks are avoided.

Modelling method, simulation and experimental verification of hypoid gear involved tooth surface deviation under manufacturing process

Tooth surface errors are inevitable in manufacturing process, and directly affect the meshing performance and NVH properties. In this paper, the tooth profile deviation and midpoint tooth pitch error were considered. Firstly, the theoretical tooth surface model was established by simulating the manufacturing process of face-hobbed hypoid gear. Then, a novel modelling method of tooth surface errors was proposed based on the relationship between discrete tooth surface points and measured error points. A solid model and finite element model by using CAD and CAE technologies were developed to study the effects of tooth surface errors and meshing phase on the contact pattern and transmission error. Results show that the gear tooth surface error was the primary factor while the tooth meshing process contributes the least of total transmission error. The peak-to-peak of the transmission error values of pinion and gear with error was 24.786 times higher than standard. The meshing phase of gear had greater impact on the contact pattern. The range of transmission error fluctuation was 210 μrad within different meshing phases. The tooth surface points measurement and rolling experiment were performed and the fluctuation range was around 233.86 μrad with a 10.2% difference.

Particle tracking velocimetry as a method for chip ejection studies during groove milling of particleboard

Studies on chip ejection from tools provide important information for the design of tools and effective chip collection elements used in woodworking machines. Among the chip properties, chip motion and its distribution are of particular interest for the design process. Until now, chip velocities have only been measured by manual tracking over a high-speed image sequence, which allows only a small scope of inspection. Here, we present the use of high-speed imaging in combination with particle tracking velocimetry as a new method for the semi-automatic evaluation of the magnitudes and directions of chip velocities. The methodology was tested in groove milling of particleboard. It was found that state-of-the-art particle tracking algorithms are suitable for quantitative analysis of chip motion in high-speed images. Therefore, spatial and temporal analysis of the chip velocity along the tool circumference are feasible and are presented here. In addition to chip velocity, chip collisions with the tool or other chips can be observed. This research also shows that image evaluation of chip sizes and shapes is potentially possible. In summary, the presented work provides methods that can quantitatively describe chip motion after chip formation. The experiments indicate that with each tooth engagement, new chips are formed, which initially move into the chip space at a median velocity higher than the cutting speed. After collisions with the tool and interparticle collisions, the particles leave the chip space of the tool at lower speeds. The machining tests performed with different process settings showed differences in the analysis results of chip movement. In the future, the presented methodology offers the possibility of investigating the relationships between tooth and chip space geometries, as well as different materials and the chip ejection of tools. Thus, the presented methodology provides a basis for creating a more general understanding of chip motion from machining operations, which can lead to innovations and improvements in chip collection.