Dynamic modelling and natural characteristic analysis of cycloid ball transmission using lumped stiffness method
Zhang et al. Robot. Biomim.
Dynamic modelling and natural characteristic analysis of cycloid ball transmission using lumped stiffness method
Peng Zhang 0
Bingbing Bao 0
0 Department of Mechanical Engineering, Anhui University of Technology , Maanshan 243000 , China
The vibration of robot joint reducer is the main factor that causes vibration or motion error of robot system. To improve the dynamic precision of robot system, the cycloid ball transmission used in robot joint is selected as study object in this paper. An efficient dynamic modelling method is presented-lumped stiffness method. Based on lumped stiffness method, a translational-torsional coupling dynamics model of cycloid ball transmission system is established. Mesh stiffness variation excitation, damping of system are all intrinsically considered in the model. The dynamic equation of system is derived by means of relative displacement relationship among different components. Then, the natural frequencies and vibration modes of the derivative system are presented by solving the associated eigenvalue problem. Finally, the influence of the main structural parameters on the natural frequency of the system is analysed. The present research can provide a new idea for dynamic analysis of robot joint reducer and provide a more simplify dynamic modelling method for robot system with joint reducer.
Lumped stiffness method; Robot joint reducer; Natural frequencies; Vibration modes
The main inducement of vibration of high-speed robot is
robot joint reducer, and therefore, the dynamic research
for robot joint reducer is necessary. At present,
domestic and overseas scholars have made many deeply
research on cycloid ball planetary transmission,
including structure principle [
], engagement principle ,
mechanical property [
], and transmission accuracy
]. However, the dynamic analysis of it has rarely
been reported. This paper effectively establishes a simple
dynamic model of cycloid ball planetary transmission,
which matches with engineering practice. After that,
the characteristics of cycloid ball planetary transmission
are analysed, and some improvement measures are
presented with the purpose of reducing vibration and
providing new ideas for robot dynamic analysis.
For the moment, the dynamic models of planetary
gear mainly include purely rotational model [
translational–torsional coupling model [
]. In purely
rotational model, the component’s torsional degree of
freedom is only considered. The model is simple because
there are few factors are considered.
Translational–torsional coupling model also includes the component’s
translational degrees of freedom. Compared with purely
rotational model, translational–torsional coupling model
is more complex, and solving is more difficult.
Therefore, it is usually used in theoretical analysis. The result
of Ref. [
] shows that when the ratio of support stiffness
to mesh stiffness is greater than 10, the simplified purely
rotational model and translational–torsional coupling
model have some equivalence in the inherent
characteristics. For cycloid ball planetary transmission, the
translational–torsional coupling model is established, and the
inherent characteristic is analysed in Refs. [
the modelling methods are too complex and difficult,
especially for a large degree of freedom dynamic system.
In view of that, this paper uses the effectively and
simple modelling method—lumped stiffness method to
establish the translational–torsional coupling model of
cycloid ball planetary transmission. Then, the natural
frequencies and vibration modes are revealed by solving
dynamic equations of system with the purpose of
providing guidance for system design.
Lumped stiffness modelling
The structure of cycloid ball planetary transmission is
shown in Fig. 1. Cycloid ball engagement pairs consist of
hypocycloid groove in the left end face of planetary disc,
epicycloid groove in the right end face of centre disc, and
balls between two discs.
This paper uses cross-ball equal-speed mechanism
as output structure for the requirements of robot joint.
Cross-ball equal-speed mechanism is made up with the
horizontal taper grooves in the right end face of
planetary disc, the horizontal taper grooves in the left end
face of cross-disc, the taper grooves in the left end face
of cross-disc, the taper grooves in the right end face of
end cover disc, and balls among three discs. In this paper,
cross-ball equal-speed mechanism is proposed, and
centre disc is treated as output disc.
Lumped stiffness model
To simplify the dynamic model, an efficient dynamic
modelling method—lumped stiffness method is proposed
based on the lumped mass method. The basic thought of
lumped stiffness method is as follows: first, the total
meshing component force along axis direction will be obtained
through mechanical analysis; second, the maximum
deformation of meshing point is considered as global
deformation, and the component of global deformation along axis
direction can be presented; finally, the ratio of total
meshing component force to global component deformation
along axis direction will be obtained. Obviously, the ratio is
lumped stiffness. Compared with the traditional modelling
method, the advantages of lumped stiffness method are as
follows: nonlinear stiffness, time-varying curvature, and
time-varying load have been integrated into the lumped
stiffness model and not directly reflected in the dynamic
model; the dynamic model will be established and solved
easily. The lumped stiffness model of cycloid ball
meshing pair and cross-ball meshing pair is, respectively, solved
using lumped stiffness method.
The mechanical model of cycloid ball engagement
pairs is shown in Fig. 2. Reference [
] shows that the total
meshing force of y axis is zero, but the total meshing
force of x axis exists. Therefore, only the lumped stiffness
model of x axis is needed.
According to the mechanical model, the stiffness model
of cycloid ball meshing pairs can be established as shown
in Fig. 3. Figure 3a shows the traditional stiffness model
of cycloid ball meshing pairs, (b) shows the lumped
stiffness model of cycloid ball meshing pairs. Obviously, the
distribution of meshing force is complex. If the meshing
forces are not effectively synthesized in modelling, the
complexity of modelling will increase. Hence, lumped
stiffness model is more convenient and simple compared
to traditional stiffness model.
According to the thought of lumped stiffness method,
the lumped stiffness of x axis is
iZ=m1 Ni cos β |x
(δ1, δ2, . . . , δZm )max cos β|x
−−→ −−→ −−−→
kmδ1 + kmδ2 + · · · + kmδZm |x
(δ1, δ2, . . . , δZm )max|x
iZ=m1 kmδm sin2 θi
= a km
where θi represents the angle between the normal line of
the i meshing point and the y axis. δm is the maximum
deformation in theory, which corresponding to the
special location. δimax is the maximum deformation at any
time during the operation. km is the meshing stiffness
of single cycloid ball meshing pair. Zm is the number of
ball. β is the half angle of cycloid groove. a is deformation
coefficient, a = a′ · sin θi, a′ = δimax δm, where sin θi is
the average value of the corresponding change interval.
In addition, the torsional angular displacement of discs
in cycloid ball meshing pair is generated by meshing
displacement. More importantly, the direction of meshing
displacement and meshing force are identical. Hence,
for the convenience of calculation, the torsional angular
displacement is substituted by torsional linear
displacement along the direction of meshing force. The lumped
torsional stiffness is substituted by lumped stiffness of x
where R is the distribution circle radius of taper grooves;
φ is the angle between the straight lines formed by the
components and the cross guide rod in the equivalent
mechanism of cross-ball equal-speed mechanism; e is
eccentric distance of input shaft.
Translational–torsional coupling model
To press close to the physical reality and avoid the
complexity of mathematical treatment, the following
simplifications and assumptions are made in the dynamic
a. Balls are regarded as elastic element because of the
b. Balls are pure rolling in the grooves, and the
influence of friction force is ignored;
c. The backlash can be eliminated by clearance screw
mechanism, and the influence of backlash
nonlinearity is ignored;
d. The cross-disc is in a floating state, and the effects of
cross-disc are not counted.
For the convenience of description of the
relationship between the components of cycloid ball planetary
transmission, this paper adopts a servo reference
system of eccentric shaft (input shaft). Thus, the geometric
centre of the input shaft is the coordinate origin. The
coordinate system rotates at the speed of input shaft.
According to the force analysis, a planar problem is
considered where input shaft, centre disc, and planetary disc
have two degrees of freedom: one translational around
its own axis and one rotational along the x axis. End
cover disc has one translational degree of freedom. In
total, the model has seven degrees of freedom. Figure 6
shows the translational—torsional coupling model of
cycloid ball planetary transmission. The sequence
number of the components in Fig. 6 is consistent with the
sequence number in Fig. 1.
0 0 −k2x
0 0 k2x
Kmx Kmx −Kmx
0 0 −c2x
0 0 c2x
Cmx Cmx −Cmx
Kmx + Kw −Kw
Cmx + Cw −Cw
where X is generalized coordinate array; M is generalized
mass matrix; F is external excitation array; K b, K m, K ω
are support stiffness matrix, mesh stiffness matrix, and
centripetal stiffness matrix; Cb, Cm are support damping
matrix and meshing damping matrix. The elements Cmx
and Cw in the matrix Cm have the following form:
K m is time-varying matrix because the lumped
stiffness Kmx is a time-varying element with the parameter θi.
To solve the problem conveniently, the θi is converted to
input shaft angle and the higher-order term is omitted.
Relative displacement between components and dynamic
The relative displacement between components is clear
because the system has fewer components. The specific
contents are shown as follows:
1. Relative displacement between centre disc and
The differential equation of system can be obtained
using Newton’s second law:
m1(x¨1 − ω12x1) + k3xδ13 + c3xδ˙13 + k1xx1 + c1xx˙1 = 0
eJ12 u¨ 1 − k3xδ13x − c3xδ˙13x + k1uu1 + c1u u˙1 = Tei
m2(x¨3 − ω12x2) + Kmxδ23 + Cmxδ˙23 + k2xx2 + c2xx˙2 = 0
J22 u¨ 2 + Kmxδ23 + Cmxδ˙23 + k2uu2 + c2uu˙ 2 = − Tr2o
mJ323u¨(3x¨3−−Kωm12xxδ233) −− CKmmxxδδ˙2233 −− KCwmδx5δ˙323−−Ckw3δx˙5δ313=x−0 c3xδ˙13x = 0
rrJ5532 u¨ 5 + Kwδ53 + Cwδ˙53 + k5uu5 + c5uu˙ 5 = 0
where Ji is the moment of inertia of component i
(i = 1, 2, 3, 5); mi is the mass of component i (i = 1, 2, 3);
ri is the pitch radius of component i (i = 1, 2, 3), r3 = r5;
cix is the lateral brace damping coefficient of component
i (i = 1, 2, 3); ciu is the torsion brace damping coefficient
of component i(i = 1, 2, 5); Cmx is the meshing damping
coefficient of cycloid ball meshing pair; Cw is the
meshing damping coefficient of cross-ball meshing pair; Ti is
the input torque of input shaft; To is the input torque of
output disc(centre disc).
The formula (6) is arranged in matrix form:
MX¨ + (Cb + Cm)X˙ + (K b + K m + K ω)X = F (7)
X = [x1, u1, x2, u2, x3, u3, u5]T
M = diag m1, J1/e2, m2, J2/r22, m3, J3/r32, J5/r52
F = [0, Ti/e, 0, −To/r2, 0, 0, 0]T
K b = diag(k1x, k1u, k2x, k2u, k3x, 0, k5u)
Cb = diag(c1x, c1u, c2x, c2u, c3x, 0, c5u)
Cmx = a cm
Cw = 6cw
Kmx = a km
14], but the mathematical model of cycloid ball planetary
transmission is identical.
Natural characteristic analysis
Natural frequency and principal mode
The natural characteristic of cycloid ball planetary
transmission can be presented by solving the eigenvalue
problem of derivative system. The eigenvalue problem of
formula (11) is
(K b + K ′m + K ω)ϕi − ωi2Mϕi = 0
where ωi is the i order natural circular frequency of
system; φi is the i order principal mode of system,
ϕi = ϕ1(ix), ϕ1(iu), ϕ2(ix), ϕ2(iu), ϕ3(ix), ϕ3(iu), ϕ5(iu)
Without loss of generality, take the cycloid ball planetary
transmission used in robot joint as an example, the dynamic
characteristics are simulated and analysed. The cross ball
equal-speed mechanism is arranged in front of the cycloid
ball meshing pair in the prototype. In other words, end the
cover disc is fixed and the central disc is used as output
component. The speed of input shaft is 1000 r/min; the meshing
stiffness of single cycloid ball meshing pair is 2.87 × 107 N/m;
the meshing stiffness of single cross-ball meshing pair is
4.44 × 107 N/m, and the deformation coefficient a is 0.9978.
Other basic parameters are shown in Table 1.
By solving the formula (12), the natural frequencies and
the principal modes of the system are obtained as shown
in Table 2. All natural frequencies are single. The
firstorder natural frequency is 0, which represents the rigid
motion of system. The vibration modes corresponding to
the other six-order natural frequencies are both
translational vibration and torsional vibration. Furthermore, the
approximate results of natural frequencies and
principal modes of cycloid ball planetary transmission can be
obtained when prototype data in this paper are plugged
into the dynamic model of literature [
Parametric influence of natural frequency
It is necessary to analyse the change regulation of natural
frequency relative to parameters of system with the
purpose of avoiding vibration. In this paper, based on
translational–torsional coupling model, the natural frequency
curves of each order are obtained by calculating
eigenvalue problem with consideration of main parameters, as
shown in Fig. 7, 8, 9 and 10.
As shown in Fig. 7, when the mass of the planetary
disc is less than 2.5 kg, the fifth- and seventh-order
natural frequencies decrease sharply, and other orders are
weakly affected. When the mass of the planetary disc is
bigger than 2.5 kg, all the natural frequencies have barely
As shown in Fig. 8, all the natural frequencies
increase gradually with the increase in the eccentric,
except for the first order. When the eccentricity is
2.5 mm, mode transition appears between the
fourthand fifth-order natural frequencies. At the point of
mode transition, the subtle change in parameters will
lead to drastic change in natural frequencies. Hence,
the sensitive points of parameters should be avoided
in the design to avoid drastic change in transmission
As shown in Fig. 9, the mesh stiffness has little effect on
the first 5 orders natural frequencies of system. The
sixthorder and seventh-order natural frequencies increase
with the increase in meshing stiffness. When meshing
stiffness increases to 3 × 107 N/m, the sixth order natural
frequency remains constant, but the seventh order
natural frequency rises dramatically.
As shown in Fig. 10, the bearing support stiffness has
certain influence on the natural frequencies, except for
the first order. When the bearing support stiffness is less
than 1 × 108 N/m, the natural frequencies increase
obviously with the increase in bearing support stiffness,
especially the fifth order; when the bearing support stiffness
is greater than 1 × 108 N/m, the sixth- and seventh-order
natural frequencies increase significantly. Modal
transition phenomenon occurs in the fifth- and sixth-order
natural frequencies when bearing support stiffness is
1 × 108 N/m, which should be avoided in the
optimization design of system.
1. To improve the motion accuracy of robot system,
the cycloid steel ball planetary transmission used in
robot joint is selected as research object. An efficient
dynamic modelling method is presented—lumped
stiffness method. A translational–torsional coupling
model is modelling, and the natural characteristics of
system are revealed.
2. All natural frequencies of system are single. The
firstorder natural frequency is 0, which represents the
rigid motion of system. The vibration modes
corresponding to the other six-order natural frequencies
are both translational vibration and torsional
3. The number of eccentricity distance and bearing
support stiffness may lead to the modal transition
phenomenon. The sensitive points of parameters should
be avoided as far as possible in the optimization
design of system.
PZ created lumped stiffness method. BB established the dynamic model,
made the natural characteristics analysis, and wrote the manuscript. PZ
supervised the research. Both authors read and approved the final manuscript.
This research study was supported financially by National Natural Science
Foundation of China (Grant No. 51405003). We acknowledge and thank their
The authors declare that they have no competing interests.
Springer Nature remains neutral with regard to jurisdictional claims in
published maps and institutional affiliations.
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