# Meshing Force of Misaligned Spline Coupling and the Influence on Rotor System

Authors: Guang Zhao; Zhansheng Liu; Feng Chen
Journal: International Journal of Rotating Machinery
 (2008)

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## Abstract

Meshing force of misaligned spline coupling is derived, dynamic equation of rotor-spline coupling system is established based on finite element analysis, the influence of meshing force on rotor-spline coupling system is simulated by numerical integral method. According to the theoretical analysis, meshing force of spline coupling is related to coupling parameters, misalignment, transmitting torque, static misalignment, dynamic vibration displacement, and so on. The meshing force increases nonlinearly with increasing the spline thickness and static misalignment or decreasing alignment meshing distance (AMD). Stiffness of coupling relates to dynamic vibration displacement, and static misalignment is not a constant. Dynamic behaviors of rotor-spline coupling system reveal the following: 1X-rotating speed is the main response frequency of system when there is no misalignment; while 2X-rotating speed appears when misalignment is present. Moreover, when misalignment increases, vibration of the system gets intricate; shaft orbit departs from origin, and magnitudes of all frequencies increase. Research results can provide important criterions on both optimization design of spline coupling and trouble shooting of rotor systems.


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## Body

## 1. Introduction

The problem of misalignment
encountered in rotating machinery is of great concern to designers and
maintenance engineers. The need of understanding the phenomena and its
principle is important to practical engineers for the purpose of trouble
shooting.  Most rotating machinery consists of a driver and a driven machine
coupled through some type of couplings [1]. There are many types of
industrial couplings one of which is spline coupling. Spline coupling allows
shaft to be easily dismantled while providing a high-torque capacity for minimum size. In
addition, it can allow relative axial and radial motions between the coupled shafts [2]. For these reasons, spline coupling is one of the most commonly used couplings to link oxidizer
pump and reducing agent
pump in rocket engine, aviation engine and generator [3], reducer and
output shaft in large ship, and so on. However, because of high-rotating speed (such
as pump in rocket) or large transmitting torque (such as output shaft in ship)
of rotors connected by spline coupling, the support conditions which get much rigorous
would cause misalignment of spline coupling unavoidablely. It has been observed
by the author on several occasions that rotating machinery stability conditions
can change should the alignment state between the driver and the driven
machines changes [4]. In one side, angular misalignment can greatly
increase the slip [2], relative slip of small amplitude can occur between the
contacting tooth surfaces.
This can give rise to fretting damage which may limit the life of the
coupling and has the potential to compromise integrity. This has been a
particular concern in aircraft mechanical systems and has been studied
experimentally for many years [5, 6]. In another side, misalignment can
causes high vibrations with different symptoms that sometimes cannot be
explained [7–9].There are many discussions in the industry
regarding the stiffness, damping, and wear of spline coupling. Marmol et al. [10] developed
a mathematical model to predict the lateral and angular stiffness and damping coefficients
of spline couplings. Based on these coefficients, they calculated the natural
frequencies of a rotor system and compared them to the experimental data. With Marmol’s linear stiffness and damping coefficients, rotor-spline coupling-bearing system was studied by Park [11].
In the design charts of flanged and curvic couplings provided by
Bannister [12], flexural stiffness values were provided. In 1994, the report of NASA [13] provides the first
opportunity to quantify the angular stiffness and equivalent viscous damping
coefficients of an axial-spline
coupling used in high-speed turbomachinery with experimental method. Ku et al. [14] developed
experimental research on dynamic coefficient of spline coupling.
Peng and Li [15] considered the bending, shear, and compress deformation when
calculating the load capability of involute spline, while the deformation of elastic
foundation was neglected. Walton et al. [16] developed three nonlinear
analytical models representing interference fits and axial and curvic spline couplings.
The results show the predictability of rotor system instability due to internal
fiction, which has been experimentally observed, especially for axial splines.Summarizing the above discussions, many researchers pay attention to the
stiffness of spline coupling, experiment is an effectual and costly way. However,
because of misalignment, it is difficult to obtain the accurate results. In
addition, spline coupling, which is one element of rotor system essentially, is
separated from its rotor system, there is not enough academic research that integrates the spline
stiffness, wear, and misalignment with the rotor system. Consequently, inclusive
and principled understanding of the roots of spline wear and misalignment is difficult
to achieve.The present work studied the elastic deformation of single spline referring
to the analysis method of tooth deformation; meshing force of misaligned spline
coupling was derived according to the deformation configuration relationships
of all splines in coupling. Graphical simulations were actualized to obtain the
influence of misalignment and structure parameters on meshing force. To gain a
greater insight of the effect of spline coupling on rotor system, the dynamic
model of rotor-spline coupling system was presented with finite element
analysis, Newmark-beta numerical integration method was employed to analyzing
the dynamics of the system.


### 2. Meshing Force of Misaligned Spline Coupling

#### 2.1. Deformation of Single Spline

Meshing rigidity of gear tooth represents the synthesis effect of all the meshing teeth in the action region [17, 18], in case of sufficient
lubrication, the friction of tooth surfaces can be neglected, meshing rigidity is
mainly correlated with elastic deformation of single gear tooth [19]. Because single spline can be considered as typical tooth, the above theory is
also suitable for deriving the spline deformation. The elastic deformation of
rectangle spline can be considered as a planar problem, generally. Spline is simplified as a cantilever beam with same section in elastic
foundation according to mechanics of materials.As shown in Figure1, f is the distribution force acting on the surface of spline, Fi is its equivalent concentrate force acting on the meshing point i; the deformation caused
by Fi consists of bending, shear deformation, and deformation of elastic foundation. Because
typical spline belongs to width spline (B/H>5), deformation of elastic
foundation should not be neglected compared to bending and shear deformation.Figure 1
Deformation of single spline.The area and inertia of cross-section areA=BH  and IZc=BH3/12,
respectively; Li is the distance from i to M; thickness, width,
and height of single spline are H, B, and L, respectively; spline number is z; Poisson's ratio and elastic modulus of spline material
are v and E, respectively, according to
Cornell’s analysis [20], for width
spline, equivalent elastic modulus Ee is E/(1−v2).(1) Bending deformation is(1)δBi=FiLi33EeI.(2) Shear deformation is(2)δSi=kFiLiGA=12FiLi(1+v)5EeA.(3) Deformation of elastic foundation, refer to the
analysis of gear tooth [20], is (3)δMi=FiEe[5.306BH2Li2+2(1−v−2v2)(1−v2)BHLi+1.534B].
Thus, the total elastic deformation is (4)δi=δBi+δSi+δMi. Therefore, the rigidity of single spline is (5)Ki=Fiδ=Ee(1/3I)Li3+(5.306/BH2)Li2+[Q+X]Li+1.534/B. where Q=2(1−v−2v2)/(1−v2)BH, X=12(1+v)/5A.Equation (5) shows that the stiffness of single spline is the function of
meshing distance Li.


#### 2.2. Meshing Distance of Misaligned Spline Coupling

When there is no misalignment, the centre of male coupling is superposed with the female, as described in Figure2, the meshing distance for each spline
is the same, and we define it as alignment meshing distance (AMD)
with the value of L0.Figure 2
AMD of coupling with no misalignment.For a real system, there are two kinds of misalignments:
static misalignment(e0,φ0) and dynamic misalignment (dynamic vibration
displacement) (x, y), a general designation of these two is misalignment in the
following analysis. So the real misalignment (e,φ) is (6)e=(x+e0cosφ0)2+(y+e0sinφ0)2,cosφ=x+e0cosφ0e,sinφ=y+e0sinφ0e. For simplifying, assume that the misalignment takes place in x-axis,
that is, φ0=0 and y=0. The meshing in two couplings
is asymmetric, teeth in right parts of the two couplings meshing tightly, contrarily, the left parts meshing loosely, meshing distance for each
spline is different as presented in Figure 3.Figure 3
Meshing distance of coupling with misalignment inx-axis.Assuming the spline serial number is 1 in positive direction ofx-axis, then
it is 2,3,…,z,  anticlockwise,  the interangular from each spline to positive
direction of x-axis is (7)φi=2π(i−1)z. Accordingly,
the Li can be presented as (8)Li=L0−ecosφi.With above analysis, meshing rigidity of single spline due to
misalignment is(9)Ki=f(Li).


#### 2.3. Meshing Force of Misaligned Spline Coupling

In a real rotor-spline
coupling system, all splines meshing tightly by transmitting
large torque cause a deformation of each
spline. At the same time,  for
coupling nodes vibrating with the system, there is relative displacement
between splines in two half-couplings,
which cause another deformation of each spline; both the deformations generate
meshing force.(1) Meshing force caused by torque is as follows.When the torque transmitted
by coupling isT, (10)T=∑i=1z[FTi⋅Ri]=∑i=1z[FTi⋅(R+Li)],R is radius of the circle in
spline root, and (11)FTi=ϕLi⋅Ki,ϕ is the torsion angular for each spline caused
by deformation, as all splines constitute a whole coupling, ϕ’s for all splines are the same when coupling turns around with the static torque T. Summarizing the above
analysis, (12)T=∑i=1z[ϕLi⋅Ki⋅(R+Li)].So,ϕ is (13)ϕ=T∑i=1z[LiKi(R+Li)].Therefore, force transmitted
by each spline is(14)FTi=ϕLi⋅Ki.(2) Meshing force caused by dynamic vibration displacement is(15)Fi′=(e′sinφi)⋅Ki, and e′=x2+y2.Accordingly,
meshing force caused by misalignment is(16)Fi=FTi+F′=(ϕLi+e′sinφi)Ki.From above
equation, we know that, due to dynamic vibration displacement, meshing forces
of splines from 1 toz/2 get larger, others get smaller, and meshing force for
each spline would not be negative. Therefore, when the transmitting torque is
zero, there would be no meshing force, even dynamic vibration displacement is still present. Therefore, (17)Fi={(ϕLi+e′sinφi)Ki,ϕLi+e′sinφi>0,0,ϕLi+e′sinφi≤0.Two components of the above
force are(18)Fxi=Ficosθi,Fyi=Fisinθi,θi is angular between meshing force and positive
direction of x-axis, it is (19)θi=φi+π2.The resultant forces due to misaligned coupling are(20)Fx=∑i=1zFxi=∑i=1z[(ϕLi+e′sinφi)Kicosθi],Fy=∑i=1zFyi=∑i=1z[(ϕLi+e′sinφi)Kisinθi].


#### 2.4. Extending of Meshing Force due to Misaligned Spline Coupling

The above analysis is a special
misalignment that appears
inx-axis, when the real misalignment is present in any direction, as shown in Figure 4,
which means that
there is an angular φ from x-axis, anticlockwise. The definition of φ is the same as (6).Figure 4
Meshing distance of coupling with misalignment in any direction.The real meshing forces of misaligned coupling are(21)fx=Fxcosφ−Fysinφ,fy=Fxsinφ+Fycosφ.From above analysis, the
meshing force model of misaligned coupling is achieved; it is a function of coupling
parameters, transmitting torque, static misalignment and dynamic vibration
displacement, and so on.


### 3. Simulations of Meshing Force

Structure parameters of spline
coupling adopted in this paper are shown in Table1.Table 1
Structure parameters of spline coupling.
NameSymbolUnitValueNumber of splinesZ14Spline
lengthLmm5.07Spline
widthBmm100Spline
thicknessHmm16AMD (alignment
meshing distance)L0mm3.375Poisson's ratiov0.3Elastic modulusEPa2.12×1011Radius
of spline rootRmm69.5Transmitting torqueTNm87000


#### 3.1. Meshing Force versus Misalignment

(1) Meshing force versus static misalignment is as follows.When the dynamic vibration displacement isx=y=0.001 mm, the static misalignment e0 changes from 0 to 1 mm, and φ0 is 0° and 30°, respectively, the curves of meshing forces are shown
in Figure 5.Figure 5
Meshing forces versus static misalignment.According to Figure5, when the dynamic vibration displacement is small and keeps constant, the following hold.(a)
Two meshing forces are equal when there is no misalignment.(b)
Meshing forces change nonlinearly with smaller static misalignment, while they change linearly when
the misalignment gets larger.(2) Meshing force versus dynamic vibration displacement  is as follows.When the static misalignment
ise0=0.5 mm, φ0=90°, the dynamic vibration displacement of x and y changes from 0 to 0.02 mm, the curves of meshing forces can be obtained as shown
in Figure 6.Figure 6
Meshing forces versus dynamic vibration displacement.Figure6 shows that when the static
misalignment appears and keeps constant, the following hold.(a)
Meshing
force changes linearly with dynamic vibration displacement, approximately, but
there is an inflexion.(b)
The slope
coefficient of meshing force curve reflects the stiffness of coupling,
consequently, during the vibration of system, the stiffness of coupling is not a constant, it relates to dynamic
vibration displacement and depends on static misalignment.


#### 3.2. Meshing Force versus Coupling Parameters

Choose the misalignment asx=y=0, e0=0.5 mm, and φ0=45°,
change the alignment meshing distance (AMD) L0 from 3 to 5.0 mm, the spline thickness H from 10
to 30 mm, and the spline width B from 50 to 150 mm, respectively, the curves of
meshing force are represented as shown in Figures 7, 8, and 9.Figure 7
Meshing force versus AMD.Figure 8
Meshing force versus spline thickness.Figure 9
Meshing force versus spline width.The above simulation results show that meshing force increases nonlinearly with the increasing alignment meshing distance (AMD) and
with the decreasing spline thickness. The spline number and width have no influence on meshing force directly. However,
for a spline coupling, the thickness of spline should be changed when the
number of spline alters.


### 4. Dynamic Model of Rotor-Spline Coupling System

2-span symmetry rotor system was used to study the
influence of spline coupling on dyamics of rotor sytem, the sketch map of
system is shown in Figure10.Figure 10
Sketch map of rotor-spline coupling system.Based on finite element analysis, presumption of Euler-Bernoulli
beam was employed to derive the dynamic equation of rotor-spline coupling
system:(22)[M]{Z¨}+[C]{Z˙}+[K]{Z}={Q(t)}+{FSP}.For simplifying, the gravity was neglected and the support
was assumed as linear spring. In above equation,[M], [C], [K] are system mass, damping (ratio damping), and
stiffness matrix, respectively; {Q(t)} is the term of unbalance force, {Z}={xj,yj,−θxj,θyj} (j is the serial number of node), {FSP} is the meshing force of spline coupling, it is (23)fx=Fx(e,e′)cosφ−Fy(e,e′)sinφ,fy=Fx(e,e′)sinφ+Fy(e,e′)cosφ. In above equation, (24)e=(e0cosφ0+dx)2+(e0sinφ0+dy)2,e′=(dx)2+(dy)2,cosφ=e0cosφ0+dxe,sinφ=e0sinφ0+dye,dx and dy are relative displacements of two
half-coupling nodes.


### 5. Dynamics Analysis of Rotor-Spline Coupling System

Nowadays, the effective method to analyze (22) is numerical method. Newmark-beta stepping integration method which has high efficiency and good stability was employed to
solve the response of system in this paper.Choosing the
static misalignment (e0, φ0) as (0, 0), (0.4 mm, 45°), and (0.8 mm, 45°),
respectively, the responses of node 7 (a half-coupling located) in the speed of 5000 rpm are shown in Figures 11 and 12, 13 and 14, 15, and 16, respectively.Frequency (0, 0).
(a)(b)Figure 12
Shaft orbit (0, 0).Frequency (0.4 mm, 45°).
(a)(b)Figure 14
Shaft orbit (0.4 mm, 45°).Frequency (0.8 mm, 45°).
(a)(b)Figure 16
Shaft orbit (0.8 mm, 45°).From above analysis,  some conclusions can be achieved as
follows.(1)
When there is no
misalignment, 1X-rotating speed is the main response frequency; shaft orbit
encloses origin.(2)
When there is
misalignment, 1X- and 2X-rotating speed are the main response frequencies.(3)
With appearance and
increases of misalignment, shaft orbit departs from origin, magnitudes of all frequencies
increase, and frequency of 2X-rotating speed increases most rapidly.


### 6. Conclusions

In this study, meshing force of misaligned spline coupling and its influence on rotor
system are studied. The followings are the outcome of the study.(1) Meshing
force of misaligned spline coupling is not related to spline number and width
directly, it changes linearly with transmitting torque while nonlinearly with alignment
meshing distance (AMD), spline thickness,
and static misalignment.(2)
Stiffness of coupling is not a constant; it relates to dynamic vibration
displacement and depends on static misalignment.(3) The
dynamics of rotor-spline coupling system shows the following: 1X-rotating speed is the main response frequency when there
is no misalignment; while 2X-rotating speed appears when misalignment is present; with the
increase of misalignment, shaft orbit departs from origin, the magnitudes of all frequencies increase,
and    2X-rotating speed increases rapidly. Thus, misalignment makes the vibration
of system more complicated.(4) In order to actualize
optimization design of spline coupling with the purpose of reducing the meshing
force caused by misalignment, reducing the transmitting torque and misalignment
are effectual ways. Moreover, decreasing spline thickness (the number of
splines should increase homologically) or increasing the alignment meshing distance (AMD) is also suitable.(5) During the trouble shooting to
rotor-spline coupling system with the aim of decreasing the rotor vibration and
reducing frequency components, the effective operations are aligning the support,
monitoring the shaft orbit,
and observing the magnitudes of all frequencies especially the 2X-rotating speed which is considered to be a characteristic commonly observed in the field of misaligned rotating machinery.


## 2. Meshing Force of Misaligned Spline Coupling

### 2.1. Deformation of Single Spline

Meshing rigidity of gear tooth represents the synthesis effect of all the meshing teeth in the action region [17, 18], in case of sufficient
lubrication, the friction of tooth surfaces can be neglected, meshing rigidity is
mainly correlated with elastic deformation of single gear tooth [19]. Because single spline can be considered as typical tooth, the above theory is
also suitable for deriving the spline deformation. The elastic deformation of
rectangle spline can be considered as a planar problem, generally. Spline is simplified as a cantilever beam with same section in elastic
foundation according to mechanics of materials.As shown in Figure1, f is the distribution force acting on the surface of spline, Fi is its equivalent concentrate force acting on the meshing point i; the deformation caused
by Fi consists of bending, shear deformation, and deformation of elastic foundation. Because
typical spline belongs to width spline (B/H>5), deformation of elastic
foundation should not be neglected compared to bending and shear deformation.Figure 1
Deformation of single spline.The area and inertia of cross-section areA=BH  and IZc=BH3/12,
respectively; Li is the distance from i to M; thickness, width,
and height of single spline are H, B, and L, respectively; spline number is z; Poisson's ratio and elastic modulus of spline material
are v and E, respectively, according to
Cornell’s analysis [20], for width
spline, equivalent elastic modulus Ee is E/(1−v2).(1) Bending deformation is(1)δBi=FiLi33EeI.(2) Shear deformation is(2)δSi=kFiLiGA=12FiLi(1+v)5EeA.(3) Deformation of elastic foundation, refer to the
analysis of gear tooth [20], is (3)δMi=FiEe[5.306BH2Li2+2(1−v−2v2)(1−v2)BHLi+1.534B].
Thus, the total elastic deformation is (4)δi=δBi+δSi+δMi. Therefore, the rigidity of single spline is (5)Ki=Fiδ=Ee(1/3I)Li3+(5.306/BH2)Li2+[Q+X]Li+1.534/B. where Q=2(1−v−2v2)/(1−v2)BH, X=12(1+v)/5A.Equation (5) shows that the stiffness of single spline is the function of
meshing distance Li.


### 2.2. Meshing Distance of Misaligned Spline Coupling

When there is no misalignment, the centre of male coupling is superposed with the female, as described in Figure2, the meshing distance for each spline
is the same, and we define it as alignment meshing distance (AMD)
with the value of L0.Figure 2
AMD of coupling with no misalignment.For a real system, there are two kinds of misalignments:
static misalignment(e0,φ0) and dynamic misalignment (dynamic vibration
displacement) (x, y), a general designation of these two is misalignment in the
following analysis. So the real misalignment (e,φ) is (6)e=(x+e0cosφ0)2+(y+e0sinφ0)2,cosφ=x+e0cosφ0e,sinφ=y+e0sinφ0e. For simplifying, assume that the misalignment takes place in x-axis,
that is, φ0=0 and y=0. The meshing in two couplings
is asymmetric, teeth in right parts of the two couplings meshing tightly, contrarily, the left parts meshing loosely, meshing distance for each
spline is different as presented in Figure 3.Figure 3
Meshing distance of coupling with misalignment inx-axis.Assuming the spline serial number is 1 in positive direction ofx-axis, then
it is 2,3,…,z,  anticlockwise,  the interangular from each spline to positive
direction of x-axis is (7)φi=2π(i−1)z. Accordingly,
the Li can be presented as (8)Li=L0−ecosφi.With above analysis, meshing rigidity of single spline due to
misalignment is(9)Ki=f(Li).


### 2.3. Meshing Force of Misaligned Spline Coupling

In a real rotor-spline
coupling system, all splines meshing tightly by transmitting
large torque cause a deformation of each
spline. At the same time,  for
coupling nodes vibrating with the system, there is relative displacement
between splines in two half-couplings,
which cause another deformation of each spline; both the deformations generate
meshing force.(1) Meshing force caused by torque is as follows.When the torque transmitted
by coupling isT, (10)T=∑i=1z[FTi⋅Ri]=∑i=1z[FTi⋅(R+Li)],R is radius of the circle in
spline root, and (11)FTi=ϕLi⋅Ki,ϕ is the torsion angular for each spline caused
by deformation, as all splines constitute a whole coupling, ϕ’s for all splines are the same when coupling turns around with the static torque T. Summarizing the above
analysis, (12)T=∑i=1z[ϕLi⋅Ki⋅(R+Li)].So,ϕ is (13)ϕ=T∑i=1z[LiKi(R+Li)].Therefore, force transmitted
by each spline is(14)FTi=ϕLi⋅Ki.(2) Meshing force caused by dynamic vibration displacement is(15)Fi′=(e′sinφi)⋅Ki, and e′=x2+y2.Accordingly,
meshing force caused by misalignment is(16)Fi=FTi+F′=(ϕLi+e′sinφi)Ki.From above
equation, we know that, due to dynamic vibration displacement, meshing forces
of splines from 1 toz/2 get larger, others get smaller, and meshing force for
each spline would not be negative. Therefore, when the transmitting torque is
zero, there would be no meshing force, even dynamic vibration displacement is still present. Therefore, (17)Fi={(ϕLi+e′sinφi)Ki,ϕLi+e′sinφi>0,0,ϕLi+e′sinφi≤0.Two components of the above
force are(18)Fxi=Ficosθi,Fyi=Fisinθi,θi is angular between meshing force and positive
direction of x-axis, it is (19)θi=φi+π2.The resultant forces due to misaligned coupling are(20)Fx=∑i=1zFxi=∑i=1z[(ϕLi+e′sinφi)Kicosθi],Fy=∑i=1zFyi=∑i=1z[(ϕLi+e′sinφi)Kisinθi].


### 2.4. Extending of Meshing Force due to Misaligned Spline Coupling

The above analysis is a special
misalignment that appears
inx-axis, when the real misalignment is present in any direction, as shown in Figure 4,
which means that
there is an angular φ from x-axis, anticlockwise. The definition of φ is the same as (6).Figure 4
Meshing distance of coupling with misalignment in any direction.The real meshing forces of misaligned coupling are(21)fx=Fxcosφ−Fysinφ,fy=Fxsinφ+Fycosφ.From above analysis, the
meshing force model of misaligned coupling is achieved; it is a function of coupling
parameters, transmitting torque, static misalignment and dynamic vibration
displacement, and so on.


## 2.1. Deformation of Single Spline

Meshing rigidity of gear tooth represents the synthesis effect of all the meshing teeth in the action region [17, 18], in case of sufficient
lubrication, the friction of tooth surfaces can be neglected, meshing rigidity is
mainly correlated with elastic deformation of single gear tooth [19]. Because single spline can be considered as typical tooth, the above theory is
also suitable for deriving the spline deformation. The elastic deformation of
rectangle spline can be considered as a planar problem, generally. Spline is simplified as a cantilever beam with same section in elastic
foundation according to mechanics of materials.As shown in Figure1, f is the distribution force acting on the surface of spline, Fi is its equivalent concentrate force acting on the meshing point i; the deformation caused
by Fi consists of bending, shear deformation, and deformation of elastic foundation. Because
typical spline belongs to width spline (B/H>5), deformation of elastic
foundation should not be neglected compared to bending and shear deformation.Figure 1
Deformation of single spline.The area and inertia of cross-section areA=BH  and IZc=BH3/12,
respectively; Li is the distance from i to M; thickness, width,
and height of single spline are H, B, and L, respectively; spline number is z; Poisson's ratio and elastic modulus of spline material
are v and E, respectively, according to
Cornell’s analysis [20], for width
spline, equivalent elastic modulus Ee is E/(1−v2).(1) Bending deformation is(1)δBi=FiLi33EeI.(2) Shear deformation is(2)δSi=kFiLiGA=12FiLi(1+v)5EeA.(3) Deformation of elastic foundation, refer to the
analysis of gear tooth [20], is (3)δMi=FiEe[5.306BH2Li2+2(1−v−2v2)(1−v2)BHLi+1.534B].
Thus, the total elastic deformation is (4)δi=δBi+δSi+δMi. Therefore, the rigidity of single spline is (5)Ki=Fiδ=Ee(1/3I)Li3+(5.306/BH2)Li2+[Q+X]Li+1.534/B. where Q=2(1−v−2v2)/(1−v2)BH, X=12(1+v)/5A.Equation (5) shows that the stiffness of single spline is the function of
meshing distance Li.


## 2.2. Meshing Distance of Misaligned Spline Coupling

When there is no misalignment, the centre of male coupling is superposed with the female, as described in Figure2, the meshing distance for each spline
is the same, and we define it as alignment meshing distance (AMD)
with the value of L0.Figure 2
AMD of coupling with no misalignment.For a real system, there are two kinds of misalignments:
static misalignment(e0,φ0) and dynamic misalignment (dynamic vibration
displacement) (x, y), a general designation of these two is misalignment in the
following analysis. So the real misalignment (e,φ) is (6)e=(x+e0cosφ0)2+(y+e0sinφ0)2,cosφ=x+e0cosφ0e,sinφ=y+e0sinφ0e. For simplifying, assume that the misalignment takes place in x-axis,
that is, φ0=0 and y=0. The meshing in two couplings
is asymmetric, teeth in right parts of the two couplings meshing tightly, contrarily, the left parts meshing loosely, meshing distance for each
spline is different as presented in Figure 3.Figure 3
Meshing distance of coupling with misalignment inx-axis.Assuming the spline serial number is 1 in positive direction ofx-axis, then
it is 2,3,…,z,  anticlockwise,  the interangular from each spline to positive
direction of x-axis is (7)φi=2π(i−1)z. Accordingly,
the Li can be presented as (8)Li=L0−ecosφi.With above analysis, meshing rigidity of single spline due to
misalignment is(9)Ki=f(Li).


## 2.3. Meshing Force of Misaligned Spline Coupling

In a real rotor-spline
coupling system, all splines meshing tightly by transmitting
large torque cause a deformation of each
spline. At the same time,  for
coupling nodes vibrating with the system, there is relative displacement
between splines in two half-couplings,
which cause another deformation of each spline; both the deformations generate
meshing force.(1) Meshing force caused by torque is as follows.When the torque transmitted
by coupling isT, (10)T=∑i=1z[FTi⋅Ri]=∑i=1z[FTi⋅(R+Li)],R is radius of the circle in
spline root, and (11)FTi=ϕLi⋅Ki,ϕ is the torsion angular for each spline caused
by deformation, as all splines constitute a whole coupling, ϕ’s for all splines are the same when coupling turns around with the static torque T. Summarizing the above
analysis, (12)T=∑i=1z[ϕLi⋅Ki⋅(R+Li)].So,ϕ is (13)ϕ=T∑i=1z[LiKi(R+Li)].Therefore, force transmitted
by each spline is(14)FTi=ϕLi⋅Ki.(2) Meshing force caused by dynamic vibration displacement is(15)Fi′=(e′sinφi)⋅Ki, and e′=x2+y2.Accordingly,
meshing force caused by misalignment is(16)Fi=FTi+F′=(ϕLi+e′sinφi)Ki.From above
equation, we know that, due to dynamic vibration displacement, meshing forces
of splines from 1 toz/2 get larger, others get smaller, and meshing force for
each spline would not be negative. Therefore, when the transmitting torque is
zero, there would be no meshing force, even dynamic vibration displacement is still present. Therefore, (17)Fi={(ϕLi+e′sinφi)Ki,ϕLi+e′sinφi>0,0,ϕLi+e′sinφi≤0.Two components of the above
force are(18)Fxi=Ficosθi,Fyi=Fisinθi,θi is angular between meshing force and positive
direction of x-axis, it is (19)θi=φi+π2.The resultant forces due to misaligned coupling are(20)Fx=∑i=1zFxi=∑i=1z[(ϕLi+e′sinφi)Kicosθi],Fy=∑i=1zFyi=∑i=1z[(ϕLi+e′sinφi)Kisinθi].


## 2.4. Extending of Meshing Force due to Misaligned Spline Coupling

The above analysis is a special
misalignment that appears
inx-axis, when the real misalignment is present in any direction, as shown in Figure 4,
which means that
there is an angular φ from x-axis, anticlockwise. The definition of φ is the same as (6).Figure 4
Meshing distance of coupling with misalignment in any direction.The real meshing forces of misaligned coupling are(21)fx=Fxcosφ−Fysinφ,fy=Fxsinφ+Fycosφ.From above analysis, the
meshing force model of misaligned coupling is achieved; it is a function of coupling
parameters, transmitting torque, static misalignment and dynamic vibration
displacement, and so on.


## 3. Simulations of Meshing Force

Structure parameters of spline
coupling adopted in this paper are shown in Table1.Table 1
Structure parameters of spline coupling.
NameSymbolUnitValueNumber of splinesZ14Spline
lengthLmm5.07Spline
widthBmm100Spline
thicknessHmm16AMD (alignment
meshing distance)L0mm3.375Poisson's ratiov0.3Elastic modulusEPa2.12×1011Radius
of spline rootRmm69.5Transmitting torqueTNm87000


### 3.1. Meshing Force versus Misalignment

(1) Meshing force versus static misalignment is as follows.When the dynamic vibration displacement isx=y=0.001 mm, the static misalignment e0 changes from 0 to 1 mm, and φ0 is 0° and 30°, respectively, the curves of meshing forces are shown
in Figure 5.Figure 5
Meshing forces versus static misalignment.According to Figure5, when the dynamic vibration displacement is small and keeps constant, the following hold.(a)
Two meshing forces are equal when there is no misalignment.(b)
Meshing forces change nonlinearly with smaller static misalignment, while they change linearly when
the misalignment gets larger.(2) Meshing force versus dynamic vibration displacement  is as follows.When the static misalignment
ise0=0.5 mm, φ0=90°, the dynamic vibration displacement of x and y changes from 0 to 0.02 mm, the curves of meshing forces can be obtained as shown
in Figure 6.Figure 6
Meshing forces versus dynamic vibration displacement.Figure6 shows that when the static
misalignment appears and keeps constant, the following hold.(a)
Meshing
force changes linearly with dynamic vibration displacement, approximately, but
there is an inflexion.(b)
The slope
coefficient of meshing force curve reflects the stiffness of coupling,
consequently, during the vibration of system, the stiffness of coupling is not a constant, it relates to dynamic
vibration displacement and depends on static misalignment.


### 3.2. Meshing Force versus Coupling Parameters

Choose the misalignment asx=y=0, e0=0.5 mm, and φ0=45°,
change the alignment meshing distance (AMD) L0 from 3 to 5.0 mm, the spline thickness H from 10
to 30 mm, and the spline width B from 50 to 150 mm, respectively, the curves of
meshing force are represented as shown in Figures 7, 8, and 9.Figure 7
Meshing force versus AMD.Figure 8
Meshing force versus spline thickness.Figure 9
Meshing force versus spline width.The above simulation results show that meshing force increases nonlinearly with the increasing alignment meshing distance (AMD) and
with the decreasing spline thickness. The spline number and width have no influence on meshing force directly. However,
for a spline coupling, the thickness of spline should be changed when the
number of spline alters.


## 3.1. Meshing Force versus Misalignment

(1) Meshing force versus static misalignment is as follows.When the dynamic vibration displacement isx=y=0.001 mm, the static misalignment e0 changes from 0 to 1 mm, and φ0 is 0° and 30°, respectively, the curves of meshing forces are shown
in Figure 5.Figure 5
Meshing forces versus static misalignment.According to Figure5, when the dynamic vibration displacement is small and keeps constant, the following hold.(a)
Two meshing forces are equal when there is no misalignment.(b)
Meshing forces change nonlinearly with smaller static misalignment, while they change linearly when
the misalignment gets larger.(2) Meshing force versus dynamic vibration displacement  is as follows.When the static misalignment
ise0=0.5 mm, φ0=90°, the dynamic vibration displacement of x and y changes from 0 to 0.02 mm, the curves of meshing forces can be obtained as shown
in Figure 6.Figure 6
Meshing forces versus dynamic vibration displacement.Figure6 shows that when the static
misalignment appears and keeps constant, the following hold.(a)
Meshing
force changes linearly with dynamic vibration displacement, approximately, but
there is an inflexion.(b)
The slope
coefficient of meshing force curve reflects the stiffness of coupling,
consequently, during the vibration of system, the stiffness of coupling is not a constant, it relates to dynamic
vibration displacement and depends on static misalignment.


## 3.2. Meshing Force versus Coupling Parameters

Choose the misalignment asx=y=0, e0=0.5 mm, and φ0=45°,
change the alignment meshing distance (AMD) L0 from 3 to 5.0 mm, the spline thickness H from 10
to 30 mm, and the spline width B from 50 to 150 mm, respectively, the curves of
meshing force are represented as shown in Figures 7, 8, and 9.Figure 7
Meshing force versus AMD.Figure 8
Meshing force versus spline thickness.Figure 9
Meshing force versus spline width.The above simulation results show that meshing force increases nonlinearly with the increasing alignment meshing distance (AMD) and
with the decreasing spline thickness. The spline number and width have no influence on meshing force directly. However,
for a spline coupling, the thickness of spline should be changed when the
number of spline alters.


## 4. Dynamic Model of Rotor-Spline Coupling System

2-span symmetry rotor system was used to study the
influence of spline coupling on dyamics of rotor sytem, the sketch map of
system is shown in Figure10.Figure 10
Sketch map of rotor-spline coupling system.Based on finite element analysis, presumption of Euler-Bernoulli
beam was employed to derive the dynamic equation of rotor-spline coupling
system:(22)[M]{Z¨}+[C]{Z˙}+[K]{Z}={Q(t)}+{FSP}.For simplifying, the gravity was neglected and the support
was assumed as linear spring. In above equation,[M], [C], [K] are system mass, damping (ratio damping), and
stiffness matrix, respectively; {Q(t)} is the term of unbalance force, {Z}={xj,yj,−θxj,θyj} (j is the serial number of node), {FSP} is the meshing force of spline coupling, it is (23)fx=Fx(e,e′)cosφ−Fy(e,e′)sinφ,fy=Fx(e,e′)sinφ+Fy(e,e′)cosφ. In above equation, (24)e=(e0cosφ0+dx)2+(e0sinφ0+dy)2,e′=(dx)2+(dy)2,cosφ=e0cosφ0+dxe,sinφ=e0sinφ0+dye,dx and dy are relative displacements of two
half-coupling nodes.


## 5. Dynamics Analysis of Rotor-Spline Coupling System

Nowadays, the effective method to analyze (22) is numerical method. Newmark-beta stepping integration method which has high efficiency and good stability was employed to
solve the response of system in this paper.Choosing the
static misalignment (e0, φ0) as (0, 0), (0.4 mm, 45°), and (0.8 mm, 45°),
respectively, the responses of node 7 (a half-coupling located) in the speed of 5000 rpm are shown in Figures 11 and 12, 13 and 14, 15, and 16, respectively.Frequency (0, 0).
(a)(b)Figure 12
Shaft orbit (0, 0).Frequency (0.4 mm, 45°).
(a)(b)Figure 14
Shaft orbit (0.4 mm, 45°).Frequency (0.8 mm, 45°).
(a)(b)Figure 16
Shaft orbit (0.8 mm, 45°).From above analysis,  some conclusions can be achieved as
follows.(1)
When there is no
misalignment, 1X-rotating speed is the main response frequency; shaft orbit
encloses origin.(2)
When there is
misalignment, 1X- and 2X-rotating speed are the main response frequencies.(3)
With appearance and
increases of misalignment, shaft orbit departs from origin, magnitudes of all frequencies
increase, and frequency of 2X-rotating speed increases most rapidly.


## 6. Conclusions

In this study, meshing force of misaligned spline coupling and its influence on rotor
system are studied. The followings are the outcome of the study.(1) Meshing
force of misaligned spline coupling is not related to spline number and width
directly, it changes linearly with transmitting torque while nonlinearly with alignment
meshing distance (AMD), spline thickness,
and static misalignment.(2)
Stiffness of coupling is not a constant; it relates to dynamic vibration
displacement and depends on static misalignment.(3) The
dynamics of rotor-spline coupling system shows the following: 1X-rotating speed is the main response frequency when there
is no misalignment; while 2X-rotating speed appears when misalignment is present; with the
increase of misalignment, shaft orbit departs from origin, the magnitudes of all frequencies increase,
and    2X-rotating speed increases rapidly. Thus, misalignment makes the vibration
of system more complicated.(4) In order to actualize
optimization design of spline coupling with the purpose of reducing the meshing
force caused by misalignment, reducing the transmitting torque and misalignment
are effectual ways. Moreover, decreasing spline thickness (the number of
splines should increase homologically) or increasing the alignment meshing distance (AMD) is also suitable.(5) During the trouble shooting to
rotor-spline coupling system with the aim of decreasing the rotor vibration and
reducing frequency components, the effective operations are aligning the support,
monitoring the shaft orbit,
and observing the magnitudes of all frequencies especially the 2X-rotating speed which is considered to be a characteristic commonly observed in the field of misaligned rotating machinery.



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*Source: 321308-2008-05-25.xml*