Structural Characteristics of Rotate Vector Reducer Free Vibration

For RV reducer widely used in robots, vibration significantly affects its performance. A lumped parameter model is developed to investigate free vibration characteristics without and with gyroscopic effects. The dynamic model considers key factors affecting vibration such as involute and cycloid gear mesh stiffness, crankshaft bending stiffness, and bearing stiffness. For both nongyroscopic and gyroscopic systems, free vibrations are examined and compared with each other. Results reveal the specific structure of vibration modes for both systems, which results from symmetry structure of RV reducer. According to vibration of the central components, vibration modes of two systems can be classified into three types, rotational, translational, and planetary component modes. Different from nongyroscopic system, the eigenvalues with gyroscopic effects are complex-valued and speeddependent. The eigenvalue for a range of carrier speeds is obtained by numerical simulation. Divergence and flutter instability is observed at speeds adjacent to critical speeds. Furthermore, the work studies effects of key factors, which include crankshaft eccentricity and the number of pins, on eigenvalues. Finally, experiment is performed to verify the effectiveness of the dynamic model. The research of this paper is helpful for the analysis on free vibration and dynamic design of RV reducer.


Introduction
Rotate Vector (RV) reducer is widely used in the joints of industrial robots due to advantages such as large transmission ratio, high precision, high carrying capacity, high transmission efficiency, and stable working performance.Vibration generated by RV reducer significantly affects the positioning and repeatability of industrial robots, especially for high performance applications.
RV reducer is a two-stage planetary gear train which joins involute gears and cycloid gears together.The vibration of planetary gears has been extensively studied.Free vibration characteristics of single-stage planetary gears [1][2][3] and compound planetary gears [4][5][6] were studied which revealed that unique vibration structure exists in planetary gears.Lin and Parker [7] and Ericson and Parker [8] investigated natural frequency veering of planetary gears.Besides general involute planetary gears, scholars have also researched dynamics of cycloidal planetary gears.Blagojevic et al. [9] and Pascale et al. [10] developed dynamic model of a single stage cycloid drive.Zhang et al. [11,12] formulated a dynamic model of RV reducer which considered mesh stiffness and bearing stiffness.Hsieh [13] studied the dynamic contact and collision conditions of the transmission components in cycloid drives with pinwheel and nonpinwheel designs.The results of [14] showed that structural vibration modes still exist in RV reducer.Gyroscopic effect is neglected in above research of free vibration of planetary gear train.
In recent years, a few studies have included gyroscopic effect on free vibration of planetary gears.Lin and Parker [25] used eigenvalue derivatives to show the effect of carrier rotation on eigenvalue loci.Abousleiman et al. [26] (1) (2)  numerically investigated the effect of carrier rotation on the instantaneous tooth mesh stiffness and dynamic response of planetary gears.They concluded gyroscopic effects should be taken into consideration in high-speed applications.However, the gyroscopic effects are demonstrated by numerical simulation in above articles.Cooley and Parker [27] analytically investigated the modal vibration structure of highspeed planetary gears with gyroscopic effects.Moreover, they also demonstrated unusual gyroscopic system eigenvalue behavior observed in planetary gears [28] and studied the stability of high-speed planetary gears with gyroscopic effects [29].
Despite research above, systematic investigation on free vibration of RV reducer with and without gyroscopic effect has not been considered.The objective of this paper is to explore the structure of natural frequencies and vibration modes and study the effect of key factors on natural frequencies for gyroscopic system.To designers of RV reducer, the prediction of natural frequencies and vibration modes allows them to choose appropriate parameters and avoid resonance conditions when designing this planetary gear system.

Dynamic Model and Equation
RV reducer is a two-stage closed planetary gear train, which is characterized by connecting involute gears and cycloid gears as shown in Figure 1.The high-speed stage is a K-H type differential planetary gear train, which consists of sun 1, planets 2, and the output wheel 7. The low-speed stage is a K-H-V type planetary gear train, which consists of crankshafts 3, cycloid gears 4, needle wheel 5, and carrier 6.The carrier and output wheel are fixed by bolts as one component.A support bearing 9 is installed in the rotation axle of the crankshaft and a turning arm bearing 8 is installed in the junction between the crankshaft and the cycloid gear.A main bearing 10 is installed between the output wheel and the shell of the reducer.
For RV reducer, the planets are of cyclic symmetry.Besides, the crankshafts are in phase, because they are arranged in identical bias direction.The cycloid gears are also in phase due to the arrangement of crankshafts.So RV reducer has special symmetry structure.
A lumped parameter model is shown in Figure 2.Each of the sun, M planets, M crankshafts, N cycloid gears, and the output wheel is treated as rigid bodies.The pins are fixed and the influence on vibration is negligible.Component flexibility, bearings, and gear meshes are represented by linear springs.The supports of the components are modeled as two perpendicular springs with equal stiffness.The transverse stiffness of the sun, planets, crankshafts, cycloid gears, and output wheel is designated as  s ,  a ,  Hb ,  cb , and  o .The torsional stiffness of the sun, planet, and output wheel is represented as  st ,  H , and  ot .The sun-planet and cycloid-pin mesh stiffness is  s ( = 1, . . ., ) and  b ( = 1, . . ., ).
Each component has three degrees of freedom, two translations and one rotation.Translational coordinates xs , ŷs , xo , and ŷo are assigned to the sun and output wheel.The hats denote that the matrices and vectors contain dimensional variables.Translational coordinates xp , ŷp , xH , ŷH , xc , and ŷc are assigned to planet i, crankshaft i, and cycloid gear j.All rotational coordinates are chosen to be .This is illustrated in Figure 2, where θs , θp , θH , θc , and θo are shown.
According to Newton's second law and theorem of angular momentum, the equations of motion can be derived.Taking the cycloid gear as an example, Figure 2 where  Hc and  Hc are the relative displacement of the th crankshaft and the jth cycloid gear and  c is the deflection of the th cycloid-pin mesh spring. Hc = xH −  θH sin  The equations of motion for the sun, planets, crankshafts, and output wheel can be obtained in a similar manner.Assembling the system equations, the governing equations of motion can be written in matrix form as where the matrices M, Ĝ, Kb , Km , KΩ , q, and F() are given in the Appendix of [14].
The influence of external load on free vibration is small and neglected.So the associated free vibration equation is M q + Ωo Ĝ q + ( Kb + Km − Ω2 o KΩ ) q = 0. (3)

Natural Frequencies and Vibration Modes without Gyroscopic Effect
To determine the nature frequencies and vibration modes for nongyroscopic system, the time-invariant system is considered.The gyroscopic effect is neglected.The associated eigenvalue problem of ( 3) is where   ( = 1, . . ., 6 + 3 + 6) are natural frequencies and vibration modes have the form φ = [p s , pp1 , . . ., pp , pH1 , . . ., pH , pc1 , . . ., pc , po ] T with ps , pp , pH , pc , and po for deflections of the sun, planets, crankshafts, cycloid gears, and the output wheel.Vibration characteristics are illustrated through a numerical example with the parameters shown in Table 1.The parameters are calculated according to engineering prototype used in experiments.
Suppose there are three planets and cycloid gears.Substitute the parameters in Table 1 into matrices M, Kb , and Km .Then the mass and stiffness matrices can be defined and ( 4) can be solved by numerical simulation.Hence, nature frequencies can be obtained with  = 3 and  = 3.
Similarly, natural frequencies can be calculated in other cases.
All results are shown in Table 2.
As shown in Table 2, all frequencies are real-valued and structural.Numerical results also reveal the specific structure of vibration modes.Because of the symmetry structure, all vibration modes can be classified into one of three types, rotational modes, translational modes, and planetary component modes.The three vibration modes are defined by the vibration of the central components (the sun and output wheel).They are possible for gyroscopic and nongyroscopic systems.The detailed proofs of the specific structure are presented in [14].Three vibration modes have the following characteristics.
When the number of crankshaft or cycloid gear is 2, the rotational and translational modes still exist.However, the multiplicity of natural frequency associated with translational mode is one.So only the first three orders of natural frequency with  = 3 and  = 2 are listed in Table 5.
Using ( 8) in (3) gives where the matrices M, G, K b , K m , K Ω , and q are given in Appendix.Substitution of the separable solution q =   into the homogeneous form of (9) gives the polynomial eigenvalue problem where  = [p s , p p1 , . . ., p p , p H1 , . . ., p H , p c1 , . . ., p c , p o ] T is the eigenvector with gyroscopic effect and p s , p p , p H , p c , and p o are deflections of the sun, planets, crankshafts, cycloid gears, and the output wheel with gyroscopic effect.Equation ( 10) is cast into the state space form as follows: where u = [ q , q] T .A = [ M 0 0 K ] and B = [ Ω o G K −K 0 ] are the state matrix operators.
Substitution of the separable solution u =   into (11) gives the gyroscopic state eigenvalue problem: where  = [Φ, Φ] T .The eigenvalues can be calculated by solving (12) with parameters in Table 1.The eigenvalues are complex-valued and natural frequencies are defined as the imaginary part.Natural frequencies and their multiplicities with different number of crankshafts and cycloid gears are shown in Table 3. Numerical results of vibration modes reveal that all vibration modes for RV reducers can be classified into one of three types.The symmetry structure of RV reducer leads to structural characteristics of vibration modes.
Figure 3 shows representative modes shapes for RV reducer with four planets and three cycloid gears.The following paragraphs summarize the modal characteristics obtained from numerical results.
Rotational Modes.There are exactly eleven natural frequencies, each associated with rotational mode of multiplicity one.In a rotational mode, all central components have pure rotation and no translation and all planetary components (the planets, the crankshafts, and the cycloid gears) have identical motion.These characteristics are consistent with nongyroscopic system.In nongyroscopic systems, the eigenvectors are real and the motions are either in-phase (where two motions have the same sign) or 180 deg out of phase (where two motions have opposite sign).However, the eigenvectors of gyroscopic systems are complex which physically means a phase difference exists between the motions of different coordinates.Hence, there are eleven pairs of complex conjugate rotational modes of gyroscopic systems.
Translational Modes.For nongyroscopic system, there are eleven pairs of real eigensolutions, where each pair has an associated natural frequency of multiplicity two.All central components of translational modes have pure translational motion and no rotational motion.For gyroscopic system, there are exactly twenty-two pairs of complex conjugate translational modes.The translational mode eigenvalues are

Influence Factor Analysis.
The speed of the carrier is defined as critical speed Ω crit where one or more of the eigenvalues are zero.The critical speed is determined by the equation as follows: Equation ( 16) represents an eigenvalue problem where Ω 2 crit is the eigenvalue. will be called the critical speed eigenvector.From the matrix symmetries, the eigenvalues Ω 2 crit and the corresponding critical speed eigenvectors  are real-valued.
Gyroscopic systems have specific characteristics of the eigenvalues and eigenvectors.The eigenvalue for a range of nondimensional carrier speeds that includes critical speeds are calculated with parameters in Table 1.The eigenvalue loci for three modes of RV reducer are shown in Section 4.2.1 to illustrate the unusual eigenvalue behavior.

Influence of Carrier Speed.
Figure 4 shows the eigenvalue loci for rotational modes of RV reducer with three planets and four cycloid gears.Eigenvalues with Im() < 10 are shown, while higher eigenvalues have similar behavior.Rotational modes have exactly eleven eigenvalues.All eigenvalues decrease with the speed of carrier. 1 ,  2 ,  4 ,  5 , and  7 decrease rapidly to zero with carrier speed for low speed. 10 decreases slowly for low carrier speed and rapidly to zero for high carrier speed. 8 and  9 decrease slowly as carrier speed increases. 3 ,  6 , and  11 decrease rapidly for low carrier speed and slowly for high speed.Rotational modes have six critical speeds at Ω o = 0.7, 0.32, 0.63, 0.99, 2.11, 6.97.When the eigenvalues  1 ,  2 ,  4 ,  5 ,  7 , and  10 are above critical speeds, the imaginary parts are zero and the real parts are positive.It means that the eigenvalues undergo divergence instability.For Ω o < 0.17, all eigenvalues are purely imaginary and stable.
Figure 5 shows the eigenvalue loci for translational modes.Figure 6 shows the low-speed region indicated by the dashed box in Figure 5.The natural frequencies of translational mode are degenerate at zero carrier speed and split for nonzero carrier speed, which is typical behavior of gyroscopic system.Translational modes have twelve critical speeds. 1 decreases to zero at Ω o = 0.38, where the critical speed occurs.Stability is not lost at this speed because the eigenvalue remains purely imaginary.As the carrier speed increases from Ω o = 0.38,  1 increases and collides with  2 at Ω o = 0.62, where the imaginary parts and the real parts are both positive.It means that the eigenvalues experience flutter instability.For 0.62 < Ω o < 4.05,  1 and  2 increase with the carrier speed and split at Ω o = 4.05, where the real parts of  1 and  2 vanish.The stability is recovered.Similar behavior appears to  11 and  12 . 3 and  4 decrease rapidly to zero at Ω o = 0.26, where the critical speed occurs.For Ω o > 0.26, the eigenvalues are real and positive which means that divergence instability occurs.Hence, both flutter and divergence instability occur for translational mode.
Figure 7 shows the eigenvalue loci for planetary component modes. 1 and  2 are the eigenvalues that experience critical speeds. 1 decreased slowly for Ω o < 0.21, but it decreases rapidly for Ω o > 0.21. 2 decreases rapidly to zero. 3 decreases rapidly for Ω o < 0.23, but it decreases slowly for Ω o > 0.23.Planetary component modes have two critical speeds at Ω o = 0.32, 0.33.For Ω o > 0.32 and Ω o > 0.33,  1 and  2 are, respectively, real and positive which means that divergence instability occurs.

Influence of Crankshaft Eccentricity.
To investigate the influence of crankshaft eccentricity on eigenvalue, Figure 8 shows the eigenvalue loci over a range of nondimensional carrier speeds when eccentricity is 0.9 mm.The eigenvalues which are obviously affected are shown by dashed lines.Compared with Figure 4, the values of  6 for low speed are  much smaller and the overall value of  9 is smaller.Compared with Figure 5, the values of  15 and  16 associated with translational mode are smaller and the two eigenvalues have the trend to converge.However, the eigenvalues associated with planetary component mode remain unchanged.Hence, eccentricity only affects eigenvalues of rotational and translational modes.

Influence of Number of Pins.
Figure 9 shows the eigenvalue loci with different number of pins for 0 < Ω o < 1.5.The eigenvalue loci with forty and forty-five pins are, respectively, by blue and red color.As seen in the picture, the overall trend of the eigenvalue loci remains unchanged and the number of pins mainly affects the lower eigenvalues with carrier speed for low speed.Unlike crankshaft, number of pins has influence on eigenvalues of all three type modes.The eigenvalues change a lot with large curvature of the loci.

Experiments
The dynamic testing experiments are performed with engineering prototype.The basic parameters of prototype are shown in Table 4.The prototype has three planets and crankshafts and two cycloid gears.The dynamic testing setup is shown in Figure 10.The speed of the sun is controlled by the servomotor.The load is replaced with inertia plate.Vibration   signal is collected by accelerometers mounted on the inertia plate and disposed by LMS TEST.Lab.The experimental results are listed in Table 5 and compared to analytical results calculated by (4).Table 5 shows the first three natural frequencies obtained from dynamic testing and analytical calculation.Results of two methods are in good agreement and the maximum amplitude difference is within 5 percent, which verifies the effectiveness of the analytical dynamic model.

Conclusions
A dynamic model of general description of RV reducer has been developed to investigate free vibration characteristics and effect of key parameters on natural frequencies for gyroscopic system.The main results are the following: (2) For nongyroscopic system, the eigenvalues and eigenvectors are real.The motions of different coordinates are either in-phase or 180 deg out of phase.However, for gyroscopic system, the eigenvalues and eigenvectors are complex-valued, which physically means a phase difference exists between the motions of different coordinates.
(3) The eigenvalues of gyroscopic system are speeddependent.There are, respectively, six, twelve, and two critical speeds for rotational, translational, and planetary component modes.Divergence and flutter instability are observed at speeds adjacent to critical speeds.Crankshaft eccentricity only affects eigenvalues of rotational and translational modes.

Figure 2 :
Figure 2: Lumped parameter model of RV reducer and coordinates.

Figure 4 :
Figure 4: Imaginary and real parts of the eigenvalue for rotational modes.

Figure 5 :Figure 6 :
Figure 5: Imaginary and real parts of the eigenvalue for translational modes.

Figure 7 :
Figure 7: Imaginary and real parts of the eigenvalue for planetary component modes.

Figure 9 :
Figure 9: Influence of number of pins.

( 1 )
The natural frequencies and vibration modes have highly structured characteristics due to symmetry structure.Vibration modes for nongyroscopic and gyroscopic systems can be classified into three typical types, rotational modes, translational modes, and planetary component modes.Rotational and translational modes have pure rotation and translation of the central components.Only planetary components have motion in planetary component mode.

𝛿 H𝑖c𝑗𝑥 sin 𝜓 c𝑗 H𝑖 − 𝛿 H𝑖c𝑗𝑦 cos 𝜓 c𝑗 H𝑖 )
(b)shows a cycloid-pin mesh with mass  c , moment of inertia  c , radius of planet distribution circle center  H , and radius of cycloid gear pitch circle  c .Notation  is the equivalent pressure angle of the cycloid-pin mesh.The cycloid gear suffers support force from crankshafts and mesh force from pins.The equations are

Table 2 :
Natural frequencies (Hz) for various numbers of planets M and N cycloid gears.
Although the motions of each planetary component are still identical and in-phase with every other planetary component, the motions of different coordinates for each planet are not in-phase with each other.A gyroscopic rotational mode has the form s , p p1 , . .., p p1 , p H1 , . .., p H1 , p c1 , . .., p c1 , p o ] T .(

Table 4 :
Basic parameters of engineering prototype.

Table 5 :
Natural frequencies from experiment and analytical model.
Mode Experiment f /(Hz) Analytical model f /(Hz) Error Δ/(%) s  s cos  s sin  s  s cos  s cos  s sin  s sin 2  s  s sin  s  s cos  s  s sin  s  2  s sin  s − cos  s cos  s  p cos  s − sin  s sin  s − sin  s cos  s  p sin  s − s sin  s − s cos  s  s  p  s cos  s sin  s − p sin  s sin  s cos  s cos 2  s − p cos  s − p sin  s − p cos  s  2  sin  cos   c cos  sin  sin  cos  cos 2   c cos 2   c cos  sin   c cos 2   2 c cos 2