A discussion of the stress distribution, modeled in a dynamic simulation environment, on the rotating parts of cycloidal reducer.

The cycloidal style of speed reducer is commonly used in many industrial power transmission applications. This type of mechanism, known for its high torque density and extreme shock load capacity, incorporates a unique reduction mechanism, which is different from that of the more commonly understood involute gearing.

To recognize the technical benefits of the cycloidal reduction mechanism, one needs to understand the forces, load distribution and contact stresses associated with the reduction components within the mechanism. This type of study is also essential in design optimization processes to improve the overall performance of the reducer [1-6].

This study can be facilitated through an example of one tooth difference type cycloidal reducer with low reduction ratio. For simplicity, let us consider one disc reducer. The main rotating components of such reducer mechanism are shown in an exploded view in Figure 1. Here, a cycloidal disc with eight holes rotates on an eccentric bearing (cam). A rotation of the input shaft mounted eccentric cam generates swaying and rotational components of motion in the system. The swaying motion is a function of the amount of eccentricity in the eccentric cam. The larger the eccentricity, the lower the reduction ratio. The large amount of eccentricity results in making bigger hole design on the cycloidal disc to accommodate low speed shaft rollers as shown by equation 1:

D = d + 2 e                             (1)

where
D is the diameter of the disc holes,
d is low speed shaft’s roller diameter, and
e is the eccentricity.

Consequently, this bigger hole design reduces more material from the cycloidal disc and as a result, the disc undergoes greater stresses. The analysis of load distribution in dynamic conditions gives access to examine such stresses and their effects on the rotating parts.

There are several studies performed on static and dynamic problems of the gears. Numerical, analytical, and experimental approaches are used to investigate different gear profiles for load distribution, contact stress, and dynamic effects. Such work helps to gather vital information to define design guidelines and find ways to optimize the torque transmission. However, very few publications are available addressing the same analyses on the cycloidal tooth profiled gears.

The purpose of this paper is to investigate the load and stress distribution on the cycloid disc under dynamic as well as inertial effect using 3-dimensional finite element analysis. In dynamic conditions, the loads are applied as a function of time. Authors of this paper previously worked on the rigid body dynamic simulation of the cycloidal reducer to calculate motion related dynamic forces of the reducer [7]. Those forces were then applied to the rotating parts in a static FEA environment. The results, however, are limited by the rigid-body assumption, which cannot deduce material-based contact stiffness and consequently accurate contact stresses. This limitation is overcome in the current study by performing analysis in a dynamic FEA environment.

To the authors’ knowledge, the following studies have been done on this topic:
Malhotra and Parameswaran proposed a theoretical method to calculate contact force distribution on the cycloidal disc [8]. The analysis is based on one disc cycloidal reducer with one tooth difference. It was assumed that only half number of the low speed shaft and housing (outer) rollers participate in torque transmission at any instance. They also discuss the effect of various design parameters on forces and contact stresses using Hertzian formula. The work further addresses theoretical efficiency of the reducer considering friction. Blanche and Yang investigated backlash and torque ripple of the cycloidal drive using a mathematical model that considers machining tolerances [9]. The previously mentioned analytical (conventional) methods, however, cannot predict precise contact loading because of their inherent assumptions and simplifications.

In the last two decades, there has been ongoing research performed using finite element based numerical methods. Gamez et al. mentioned contact stress analysis in trochoidal gear pump application [10]. His work gives a comparison between analytical model and FEM analysis of contact stress. Photoelasticity technique is also elaborated to evaluate the stress. For finite element analysis, his work mainly focuses on two-dimensional quasi-static model of trochoidal tooth profiled gear pump. Li et al. used various tools, including Abaqus finite element software, to observe contact stresses in pinion-gear system by varying pressure angle [11]. The work also addresses stresses in planetary gearboxes by varying gap tolerance. The finite element model was developed using a two-dimensional, four-node bilinear element for nonlinear sliding contact solution.

Barone et al. investigated load sharing and stresses of face gears building a three-dimensional FEA model in Ansys software to simulate the effects of shaft misalignment and tooth profile modification. The results are shown by plotting graphs of load sharing, contact pressure, contact stress, and contact path against rotational angle [12]. This model implements penalty and augmented Lagrangian methods for surface-to-surface contact.

### Geometry of Cycloidal Reducer

One tooth difference cycloidal reducer can reduce the input speed up to 87:1 in a single stage. The gearbox Cyclo® CNH609-15, which is a horizontal, foot mounted, concentric shaft speed reducer of ratio 15:1, is selected for the simulation modeling and analysis [13]. The gearbox is chosen from the low reduction ratio category offered by the manufacturer. As discussed at the beginning, the low reduction ratio results in greater induced stresses in the cycloidal disc on account of reduced material content of the disc. This model would be observed for such stresses in the cycloidal disc. The reducer design is comprised of one disc mechanism along with a counterweight (as shown in Figure 1), acting as a substitute to the second disc, which is apparent in high torque transmitting reducers for dynamic balance and load sharing. This mechanism gives an advantage to focus on the contact loading behavior of only one cycloidal disc.

The epi-trochoidal tooth profile of the cycloidal disc is a vital feature in the reducer mechanism. Theoretically, for the one tooth difference reducer, all disc teeth or lobes on the profile remain in contact with subsequent ring gear housing rollers (outer rollers) and half of the rollers are considered participating in torque transmission at any instance [8]. However, the manufacturing errors and clearances for lubrication etc. impede the application of all tooth-roller contacts. The trochoidal profile in this model is developed from the manufacturing drawing which takes tooth modifications and tolerances into account.

### Modeling Description

Algor Graphics User Interface (GUI) and its finite element code are used to build and simulate the dynamic behavior of the cycloidal reducer. The torque transmitting components shown in Figure 2 are imported from the CAD model to Algor GUI. For more contact simplification, eccentric rollers of the cam are excluded and the corresponding gap is filled by raising the diameter of the cam. The dynamic FEA model is built creating 3D solid mesh of brick elements. A brick element comprises of eight nodes with only translational degrees of freedom in 3D space. This element type is preferred in the study since the solid mesh engine generates more consistent brick elements than any other type, like tetrahedral. Several attempts are made to establish a suitable mesh density that would capture exact geometry of the parts. For instance, the trochoidal profile on the cycloidal disc periphery has to be defined precisely in order to obtain correct mating of the assembled rotating parts.

“Surface-to-surface” contact is established between two interacting parts; namely, eccentric bearing (cam) and the disc, the cycloidal disc and outer rollers, and lastly, the cycloidal disc and low speed rollers. A constant nodal prescribed rotational motion of 20 or 30 revolutions per second, assumed to be generated by motor, is applied to the center point of the cam hole. In the absence of high-speed shaft, beam joints are created to transfer prescribed motion to the eccentric cam as shown in Figure 3.

These beams are defined as rigid elements with no mass density. Similarly, as described in Figure 4, rigid beam elements are also used to define the relative motion between the low speed shaft rollers. The output shaft load is assigned by placing a nodal moment that acts at the center of the beam element linkage, in the opposite direction of the rotation of low speed shaft rollers. The outer rollers are fixed in all degrees of freedoms at their inner diameters. The total time duration of the simulation is set for 360º rotation of the eccentric cam. This time duration is divided into small time increment steps to capture simulation history at every 2º rotation.

A total Lagrangian formulation method is adopted to determine the equilibrium of strains at each time increment. By this adoption, therefore, the model becomes geometrically non-linear where the stiffness and the geometry are updated at each step. If the simulation result were animated in GUI, the movements of the parts would be seen because of the geometric updating. AISI E52100 material properties are assigned to the elements of all parts. The material defined in this study is elastic, homogenous, and isotropic. The model is built with certain assumptions considering limitations of the finite element code and the simplification adopted in this study. It is assumed that the sliding motion is absent between the contacting surfaces. So the torque is transmitted from one part to another by perfect rolling contact. The modeling does not take into account the friction between rotating parts, manufacturing errors, wear, thermal and vibration effects.

### Simulation

Finite element commercial code uses surface-to-surface contact algorithm to analyze the contact forces generated by the interaction between parts. The contact algorithm involves an iterative process where the solution is derived from displacement-based convergence. The solution is reached iteratively at each defined time increment step by solving non-linear equations using full Newton-Raphson integration method [14].

This iterative process is presented graphically in Figure 5 where the thick blue colored line shows a theoretical stiffness curve. Here, three iterations are performed to calculate stiffness curves k1, k2and k3. The force, F, is considered as an appropriate solution for displacement, D, when an increment Δd at the last iteration, to obtain curve k3, is within the defined tolerance. To obtain precise approximation of the stiffness, one can define very small tolerance; however, it would increase the number of iterations and ultimately the total time to complete the simulation.

The contact between two surfaces is identified by applying contact stiffness between nodes of those surfaces. This contact stiffness is applied, which implies that the contact has occurred, at the moment when the defined contact tolerance is greater than the distance between the surfaces. For too small contact stiffness, the surface nodes are unable to experience the reaction force from the nodes of the other contacting surface, which results in surface penetration. If the contact stiffness is too high, the iterative time step may be reduced to a small value causing a convergence problem, which makes the model unstable. The contact stiffness is calculated by the software from the properties of material of the parts using equation 2:

(2)
where
K is contact stiffness,
fs is the scaling parameter,
E is the Young’s Modulus of the material,
L is the contact surface area, and
S is the volume of the element.

In dynamic analysis, stresses are estimated considering the material behavior, motion and gravity of the parts. Equation 3 of the simulation in matrix form relies on the combination of Hook’s law and Newton’s second law:

[M]{a} + [C]{v} + [K]{d} = {f}                  (3)

where
M is the mass,
a is the acceleration vector,
C is the dampening constant,
v is the velocity vector,
K is stiffness,
d is the displacement vector due to force, and
f is the force vector.

The involvement of the acceleration term allows simulating impact motion of the cycloidal disc on the outer rollers.

### Results and Discussion

Unlike the involute tooth profile gearing, where the contact load is concentrated only at a small area of the gear pair, the cycloidal reducer is well known for diffusing the load over its components. This load distribution ability of the cycloidal gear mechanism forms a high shock load absorption capacity. Further discussion in this section investigates contact load and stress distribution in dynamic conditions. The analysis is performed at full load 134.47 kN-mm moment, which acts on the low speed shaft in the direction opposite to the rotation of output shaft. The eccentric cam rotates one revolution under the prescribed displacement of 1200 rpm.

Figure 6 presents the VonMises stress contour of the disc, housing rollers and low speed shaft rollers at 54º from X-axis at time instance 0.005 sec. After analyzing simulation data history collected at 180 time instances, it is inferred that the cycloidal disc shares the significant amount of torque load with approximately five outer rollers at any given time.

The compressive force generated by the cycloidal disc on the outer rollers can be observed in the form of total reaction force at the rollers’ inner diameter nodes, which are fixed in all degrees of freedom. For simplified explanation, the total reaction force magnitude of five outer rollers is shown in Figure 7 against time. The data points of respective rollers are joined by splines with different line types. Note that the torque load is almost fully shared by these rollers from 0.004 sec to 0.006 sec.

Figure 7 indicates an important phenomenon of impact, which is often considered as a major factor in vibration study of the reducer. Each data point represents 2º rotational increment of the input shaft. The smoothness in these impact curves may be attained by more mesh refinement of the contacting surfaces and also by capturing more data in small period of time than shown in the figure.

Load distribution along the width of a gear is always an important criterion in the design consideration. In our study, for example, Figure 8 shows the contact force contour on the outer roller No. 14 at the same instance and the conditions defined by Figure 6. The graphical representation of forces can be plotted as shown in Figure 9. The even distribution of contact forces or curve smoothness could have been achieved by additional mesh refinement along the width of the cycloidal disc.

In further discussion, apart from the above-explained analysis, the following scenarios are performed to study the effect of different input speeds and loading conditions:

2. 1800 rpm input speed, 500 % shock load (5 times full load)
3. 1200 rpm input speed, 500 % shock load (5 times full load)

The results of all analyses agree with the findings of the discussed scenario:

Based on the local deformation theory, the contact stress between the disc and the outer roller has the following relationship with output torque (equation 4):

(4)
where
σc is contact stress, and
Tout is the reducer output torque.

This relation can be validated in the Figure 10. The shock load is kept constant after 0.016667 sec; the contact stress generated by the disc on the outer roller is compared with constant full load condition. It is found that the increase in the contact stresses is approximately proportional to For instance, at 0.0174s, the contact stresses for full load and 5 times full load are 195.95 N/mm2 and 457.05 N/mm2 respectively. The ratio of the two values is 2.33≈ Thus, the FEA results are in agreement with the local deformation theory. During the shock load analysis, a significant change in the load distribution is observed, as shown in Figure 12, where both conditions; Full load and Sudden load are compared in a certain period of time 0.02 sec (from 0.018 sec to 0.02 sec). Outer roller No. 8 and 9 share considerable amount of load as shown in Figure 12b against their negligible presence in Figure 12a.

### Conclusion

This paper discusses a three dimensional finite element method for the load distribution and dynamic contact analysis of low reduction ratio cycloidal reducer using commercially available Algor FEA code.

The discussion is mainly focused on the interaction between outer rollers and the cycloidal disc at different input speeds and loading conditions. This discussion is useful since the striking effect (impact) of these parts is directly related to the noise problem and also contributes in lower efficiency of the reducer. The reliability of the model is argued by validating the FEA derived contact stresses of the cycloidal disc with deformation theory.

### Future Study

These simulations and scenarios need to be validated experimentally. The study can be continued by addressing the friction and the sliding motion between the rotating parts, which were not considered in this paper. The assumption of immovable outer rollers can be relaxed in further investigation to enable the motion of the rollers about their axes.

A parametric study can be performed in the future, utilizing the simulation setup elaborated in this study. In this regard, the contact stress variation can be analyzed by changing geometric parameters such as eccentricity, trochoidal tooth profile of the cycloidal disc, pitch circle radius of ring gear housing rollers, etc.

### References

1. Blagojevic, M., et al., A New Design of a Two-Stage Cycloidal Speed Reducer. Journal of Mechanical Design, 2011. 133(8): p. 085001 (7 pp.).
2. Gorla, C., et al., “Theoretical and experimental analysis of a cycloidal speed reducer.” Journal of Mechanical Design, Transactions of the ASME, 2008. 130(11): p. 1126041-1126048.
3. Meng, Y., C. Wu, and L. Ling, “Mathematical modeling of the transmission performance of 2K-H pin cycloid planetary mechanism.” Mechanism and Machine Theory, 2007. 42(7): p. 776-790.
4. Sensinger, J.W., “Unified approach to cycloid drive profile, stress, and efficiency optimization.” Journal of Mechanical Design, Transactions of the ASME, 2010. 132(2): p. 0245031-0245035.
5. Shin, J.-H. and S.-M. Kwon, “On the lobe profile design in a cycloid reducer using instant velocity center.” Mechanism and Machine Theory, 2006. 41(5): p. 596-616.
6. Chen, B., et al., “Gear geometry of cycloid drives.” Science in China Series E: Technological Sciences, 2008. 51(5): p. 598-610.
7. Bobak T.R., Thube S.V., “The dynamic Simulation and Analysis of a Cycloidal Speed Reducer.” 11th ASME International Power Transmission and Gearing Conference, 2011. DETC2011-48494: p. 9.
8. Malhotra, S.K. and M.A. Parameswaran, “Analysis of a cycloid speed reducer.” Mechanism and Machine Theory, 1983. 18(6): p. 491-499.
9. Blanche, J.G. and D.C.H. Yang, “Cycloid drives with machining tolerances.” Journal of mechanisms, transmissions, and automation in design, 1989. 111(3): p. 337-344.
10. Gamez-Montero, P.J., et al., “Contact problems of a trochoidal-gear pump.” International Journal of Mechanical Sciences, 2006. 48(12): p. 1471-80.
11. Li, S., “Finite element analyses for contact strength and bending strength of a pair of spur gears with machining errors, assembly errors and tooth modifications.” Mechanism and Machine Theory, 2007. 42(1): p. 88-114.
12. Barone, S., L. Borgianni, and P. Forte, “Evaluation of the effect of misalignment and profile modification in face gear drive by a finite element meshing simulation.” Transactions of the ASME. Journal of Mechanical Design, 2004. 126(5): p. 916-24.
13. Sumitomo, Cyclo 6000 speed reducer. Catalog 03.601.50.007, 2010.
14. Autodesk, Algor Simulation In-product help. 2013.

Printed with permission of the copyright holder, the American Gear Manufacturers Association, 1001 N. Fairfax Street, Suite 500, Alexandria, Virginia 22314.  Statements presented in this paper are those of the authors and may not represent the position or opinion of the AMERICAN GEAR MANUFACTURERS ASSOCIATION.

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is product engineer with Sumitomo Machinery Corporation of America. Sandeep received his MS degree in mechanical engineering from Old Dominion University in 2009 specializing in FEA modeling and simulation. He has more than five years of work experience in mechanical industry, including project management and engineering applications.
is a research and development engineer for Sumitomo Drive Technologies. He has 18 years of combined experience in the power transmission industry working in areas of product engineering, research and development, technical customer service, and quality.For more information on Sumitomo products and services, visit www.sumitomodrive.com.