Stress Analysis on Internal nut of Ballscrew for Vertical
CNC Milling Machine
Thu Zar Khin1, Ei Ei Htwe2, Nyein Aye San3
Department of Mechanical Engineering
Mandalay Technological University
Abstract – This paper deals with stress analysis on internal
nut of ballscrew in CNC vertical milling machine (ANILAM
3000M). X-axis internal recirculation ballscrew and
preloading double nut with spacer one end flange type are
used in this paper. From calculations, the critical speed,
buckling load, total rigidity of the system, required torque
and power to moves the load, shaft deflection, torsion and
then service life of the ballscrew are obtained for making
useful design of the X-axis internal ballscrew. Stainless steel
(AISI 1035) is selected for double nut. The detail design
calculations of ballscrew are described. In static analysis,
stress had been calculated by using von-Mises criteria and
the percent of error were also calculated. And stress
analysis acting on the ball groove in ball nut is performed
by using SolidWorks software. The percentage difference of
von-Mises stress for (AISI 1035 Steel) material is 14.3%.
Keywords- double nut, internal recirculating ballscrew,
material, von-Mises stress, rigidity
I. INTRODUCTION
Mechanical working area is divided into two
portions in CNC. The first portion is the spindle drive
motion in the head stock and the second one is the
ballscrew drive mechanism. The feed drive system or
mechanism drive system contains a ball screw threaded
shaft, a ball nut body, balls, ball recirculation elements,
wipers, belts timing, pulley and servo motor. In this
mechanism, the motor drives the screw with torque
transmission elements, timing belts. So, the input power
of the ballscrew gets from timing belts and tooth
pulley and the output torque is to carry the load on the
working table.[3]
The mechanism of drive system is the important part
of the CNC machine. Ballscrew is the one of the
important part in the mechanism drive system.
A. ANILAM 3000 M Vertical Milling Machine
In this paper, the ANILAM 3000 M vertical type
milling machine is selected to study or research will be
conducted on this .paper.
ANILAM 3000 M Vertical Type Milling Machine
has a programmable and flexibility of CNC control
system configurations to ensure the perfect match for
machine tools builders, importers, distributors and for
retrofitters. It can perform the various types of milling
operation such as face milling, end milling, drilling and
so on. It comprises of with knee and column as fixed
parts and table with translational motion along X-axis,
saddle with translational motion along Y-axis and spindle
head with translational motion along Z-axis motion. In
this paper, the main focus is set on the design of
ballscrew of vertical type milling machine has been
designing.[1]
Photograph of ANILAM 3000 M is shown in Figure
2.
Figure.2. CNC System with Vertical Milling Machine [1]
II. BACKGROUND OF BALLSCREW AND NUT
Figure.1. Ballscrew Drive Mechanism [2]
Ballscrew is the one of the important part in the
mechanism drive system. It is a mechanical system which
capable of converting rotary motion to linear motion or
vice versa. The ballscrew drive consists of a ballscrew
and a ball nut with recirculating balls. The connection
between the screws and the nut is made by balls which
roll in the matching ball forms in screw and ball nut. The
forces transmitted are distributed over a large number of
balls, giving a comparatively low relative load per ball.
With rolling elements, the ball screw drive has a very low
friction coefficient.
Ball nuts are calssified into four types by ball
circulation method: external recirculation type nut with
return tube, internal recirculation type nut with return
caps, endcap recirculation type nut with return system
and end deflector recirculation type with return system.
The CNC machine can get higher efficiency by using
internal recirculation ballscrew .Gothic arch profile is
used to meet the requirements of high stiffness, smooth
motion and low friction. For X-axis ballscrew, double nut
with one end flange and double nut preloading with
spacer is used to minimize backlash. The internal
recirculation double nut with one end flange type is
shown in Figure 3.
The specification design data of the X-axis ballscrew
of CNC vertical milling machine is shown in Table I.
TABLE I
DESIGN SPECIFICATION OF X-AXIS BALLSCREW
Sir
Item
Unit
Symbol
Value
mm
Ls
1450
mm
Lt
1180
3
Total Length of
Ballscrew
Mounting Position
Length
Diameter of ballscrew
mm
D
32
4
Mass of ballscrew
kg
Ws
15
5
Root Diameter
mm
dr
29.2
6
Ball Circle Diameter
mm
Dpw
32.75
7
Insert Ball Diameter
mm
Dball
4
8
Bearing Torque
mm
Tb
13
9
Work piece weight
kg
W2
200
10
Table weight
kg
W1
200
11
Rotational speed
rpm
N
1500
12
Diameter of Nut
mm
DN
48
13
Width of flange
mm
1
2
10
TABLE II
SMOOTH RUNNING WITHOUT IMPACT CONDITION
Condition
Axial Load
Revolution
Loading Time
(kg)
(Rpm)
Ratio
(%)
1
100
1000
45
2
400
50
35
3
800
100
20
Figure 3.Internal Recirculation Type of Ballscrew[5]
III.
DESIGN SPECIFICATION FOR
CALCULATIONS OF X-AXIS BALLSCREW
For X-axis ballscrew design, double start, Gothic
arch profile, double nut with one end flange and fixedsupported type of bearing mounting are used.
Acceleration speed is 100 rad/s2.In motor condition,
motor diameter is 100 mm and motor length is 270mm.
In gear condition, driver pulley diameter G1 is 56mm,
thickness is 40mm and teeth are 22. Driven pulley
diameter is G2 is 112mm, thickness is 40mm and teeth
are 44. For smooth running without impact condition, fp
is 1.1.
Fixed-Supported method,
Motor Drive Torque
TM  (Ta  Tb  Td ) 
N1
N2
Where,
Ta 
Fb  l
2π  η1
n = 2 (Buckling load)
Mf = 0.692(Critical speed)
From standard dimension,
Dia;=32mm, lead=10 mm
IV DESIGN CONSIDERATION OF BALLSCREW
AND NUT
K=1300 N/µm
Ca=42.2 kN
Td 
Kp  P  l
2π
Fb  Fbm  μWt
t
t
t
N avg  N1  1  N 2  2  N 3  3
100
100
100
(1)
For double nut with preload,
P
Wt
2.8
Where, tan 
Input Power:
Total Inertia:
J  J M  J G  J Ballscrew  J load
(2)
1
 π  ρ  R 4  Lm
2g
W  l  N 
J load  t      1 
g  2π   N 2 
2
J ba;;screw
2  π  N  Ti
60000
Pi 
(7)
Where,
Where,
JM 
Ph
πDpw
2
N 
1

 Ws  R 2   1 
2g
 N2 
N 
J G  J G1  J G2  1 
 N2 
FA  Ph  103
2π  η1
Ti 
With preload,
FA  Fbm  P
2
Output Power:
2
2  π  N  To
60000
Po 
(8)
Where,
1
 π  ρ  R 4  LG2
2g
1
J G1   π  ρ  R 4  L G1
2g
J G2 
To 
Maximum Permissible Load:
Total Motor Torque:
TMa  TM  Ta'
FA  Ph  103  η2
2π
Fp  0.5Fk
(3)
Where,
Fk 
Where,
Ta'  J  α
(9)
I
nπ 2 EI
L2t
π 4
(d r )
64
Drive Power:
Pd 
Tpmax  N max
(4)
974
N p  0.8N c
Where,
Tpmax  2  TMa
Where,
N c  2.71  108 (
Efficiency (rotary to linear motion)
1  f  tan
η1 
f
1
tan
Maximum Permissible Speed:
(10)
Mf  dr
)
L2t
Deflection:
(5)
d rad 
Efficiency (linear to rotary motion)
Where,
f
1
tan
η2 
1  f  tan
P
(6)
K p PL4t
EI
W1  W2
Lt
(11)
I
π
(d r ) 4
32
L (2)
Thermal Deformation:
ΔL  K a  ΔΤ  L
(12)
3

  10 6


 F 
Fbm(1)  P1  bm 
3P 

3/2
Fbm(2)  Fbm(1)  Fbm
Angular Torsion:
θ
 C
 a
F
 bm(2)
To  102  Ls
GI
(13)
Service Life (hr):
Pretension Force:
Lh 
Pf  KS  Δl
(14)
L rpm
(17)
N avg  60
Impact Force:
Rigidity:
1
1
1
1



K KS K N K B
1.14  Vn2
F
k1α m
Where,
(15)
 5Vn2 
αm  

 4k1k h 
Where,
Ks 
AE
(Fixed-Supported)
1000L t
Vn 
0.45 Q 2 1/3
δ ao 
(
)
sinα D ball
Q
KB 
δi 
P 1/3
)
0.1Ca

9 / 10
Where,
L (1)
 C
 a
F
 bm(1)
1  μ i2
(i=1,2)
πE i
B
1
2R 1
k1 =
m1  m 2
m1m 2
Service Life (rev):

πDN
 sin θ
60
1 1
1 

A   
2  R1 R 2 
3P
δ ao
L  L(110) / 9  L( 210) / 9
2/5
4
q3/2
k
kh  
3 δ1  δ 2  A  B
P
Zsin α
K N  0.8  K(
(18)
(16)
For Insert Ball _m1, m1=𝜌v=𝜌 x
 
4 3
πr1
3
3

  10 6


For Ball Groove_m2, m2=𝜌v=𝜌AL
 4R 2 
A  πR 2   r 

3π 

software. The stress analysis is shown in the following
fig;
Friction Force:
Fn
F
R  R 22
2
 R  μ k R 2 
(19)
Where,
R
N s  Ph
2π
Von-Mises Criteria:
 σ1  σ 2 2  σ 2  σ 3 2  σ 3  σ1 2 
σ

2


Where, σ1 , σ 2 
σy 
V
1
σ y  σ Z   1
2
2
σ
1/2
(20)
 σ z   4τ 2
2
y
Fy
A
Figure 4.Stress Simulation of Ball Groove
RESULTS FOR X-AXIS BALLSCREW
By using input data and above data equations, the
result data are coming out as follow.
TABLE III
RESULTS FOR X-AXIS BALLSCREW
Parameters
Motor Torque in Normal (kg-mm)
Total Inertia (kg-mm-sec2)
Total Motor Torque (kg-mm)
Drive Power (Hp)
Efficiency (%)
Input Power (kW)
Output Power(kW)
Maximum Permissible Load (N)
Maximum Permissible Speed (rpm)
Deflection of Ballscrew (mm)
Thermal Deformation of Ballscrew(mm)
Torsion of Ballscrew (rad)
Rigidity of Ballscrew System (N/µm)
Pretension Force (N)
Service life (rev)
Serivce life(hr)
Maximum Impact Force (N)
Friction Force (N)
Von-Mises Stress (MPa)
Calculated
Results
318.58
2.34
552.58
2.28
90 , 89
0.62
0.51
52084.21
3146.18
2.38
0.08
8.12 x 10-5
101.9
11.7 x 103
2241.65x106
76637.61
853.26
129.66
15.126
VI STRESS ANALYSIS BY THE NUMERICAL
SIMULATION
As the driven shaft in the mechanism operates at
high rotational speed, forces caused by the impact
activity between the steel ball and ball track may generate
high stress.[4] By using the ball groove dimension,
fixture, forces result from the result table and material,
the stress simulation can perform in the SolidWork
The maximum von-Mises stress is 12.969 MPa. The
material (AISI 1035) yield strength is 282.685 MPa.
To get the safety result, compare the theoretical
result and simulation result of the von-Mises stress. The
following Table IV is comparison result of von-Mises
stress with error percentage.
TABLE IV
COMPARE THE RESULT OF VON-MISES STRESS WITH ERROR PERCENTAGE
Von-Mises Stress
Von-Mises Stress
Error Percentage
from Theoretical
from Simulation
Result
Result
15.126 MPa
12.969 MPa
14.3%
CONCLUSION
The stress analysis on ball groove in internal nut
of ballscrew for ANILAM 3000 M vertical milling
machine is done by the above procedure based on the
actual technical data from the machine. In the
consideration the design is consider for the internal
recirculation type ballscrew and double nut. The
efficiency, life circle, torque and load are calculated from
the known specification. The stress analysis is performed
by the ball groove in internal nut dimension and then the
fixture, maximum impact force, material selection using
the Solid Work software. Stainless steel is containing in
the ductile materials. The maximum impact force,
friction force and maximum von-Mises stress are 853.26
N, 129.66 N and 12.969 MPa acting on the ball groove in
internal nut. The percentage difference of von-Mises
stresses for (AISI 1035 Steel) material is 14.3%. As the
ballscrew is a dynamic component, the dynamic stress
analysis or vibration analysis also should be considered
for the next paper or result.
ACKNOWLEDGMENT
Firstly, the author would like to special thanks to her
parents for encouragement without any trouble
throughout her life. And then the author especially thanks
to her supervisor, Dr Ei Ei Htwe, Head of Mechanical
Engineering Department from Mandalay Technological
University, for her guidelines for this paper. And the
author also thanks to her co-supervisor, Dr Nyein Aye
San, Lecturer of Mechanical Engineering Department
from Mandalay Technological University, for her
guidelines for this paper. Then Dr Khin Maung Chin and
Dr Thein Min Htike, also thanks. The author greatly
express thanks to all persons whom will concern to
support in preparing this paper.
[1]
[2]
[3]
[4]
[5]
[6]
REFERENCES
Yin NweKhaing, Design and Stress analysis of
working table for 3-axis vertical CNC milling
machine( ANILAM 3000 M),2013,Mandalay,
Myanmar.
Win Kyaw Thu, Design of ballscrew for vertical
milling machine, 2011, Mandalay, Myanmar
EiEiHtwe, Design and Construction of CNC
milling machine, 2005, Yangon, Myanmar
Jui-Pin Hung, James Shih-Shyn Wu, Jerry Y.
Chiu, Impact failure analysis of recirculating
mechanism in ballscrew,2003, Taiwan.
www. hiwim.com.tw.
http:// www.thk.co.jp
NOMENCLATURE
A
Ca
Dpw
Dball
dr
E
F
Fp
Fbm
Fb
Fk
fp
Fa
G
g
I
J
JG1
JG2
JBallscrew
Jload
K
KS
Screw shaft cross-sectional area(mm2)
Dynamic load rating(kN)
Pitch circle diameter(mm)
Diameter of insert ball(mm)
Root diameter of ballscrew shaft(mm)
Young’s modulus (N/mm2)
Maximum impact force(N)
Maximum permissible load(kg)
Mean operating load(kg)
Axial load(kg)
Buckling load(kg)
Operation condition factor
Applied axial load(N)
Steel modulus of elasticity (8.1*104 N/mm2)
Gravity coefficient
Polar moment of inertia of the ballscrew shaft
cross section (mm4)
System inertia(kg-mm-s2)
Inertia of driver gear(kg-mm-s2)
Inertia of driven gear(kg-mm-s2)
Ineria of ballscrew(kg-mm-s2)
Inertia of load(kg-mm-s2)
Rigiditty of the screw system(N/µm)
Rigidity of the screw shaft(N/µm)
KN
KB
K
Kp
Kp
kh
k1
Lt
Lm
LG1
l
LG2
Ls
Mf
N1
N2
Nc
N
Navg
n
Np
Nmax
Pi
Po
Pf
P
P
Pd
Q
Ta
Ti
TM
Tb
Td
T a’
To
TMa
Tpmax
Ws
Wt
Z
Rigidity of the nut(N/µm)
Rigidity of the support bearing(N/µm)
Rigidity value from standard dimension(N/µm)
Deflection factor according to the type of
mounting
Preload torque coefficient(0.1~0.3)
Hertz stiffness(N/µm)
Mass constant(kg-1)
Distance between support bearing(mm)
Length of motor(mm)
Length of driver gear(mm)
lead(mm)
Length of driven gear(mm)
Total length of screw(mm)
Factor for different mounting types
Number of teeth for driver gear
Number of teeth for driven gear
Critical speed(rpm)
Speed(rpm)
Average speed(rpm)
Factor for different mounting types
Maximum permissible speed(rpm)
Maximum rotational speed(rpm)
Input power (kW)
Output power(kW)
Pretension force(kg)
Distributed weight(kg/mm)
Preload force(kg)
Maximum drive power(W)
Axial load(N)
Drive torque for common transmission (kg-mm)
Driving moment to be applied(N-mm)
Motor drive torque (kg-mm)
Friction torque of supporting bearing (kgmm)
Preload drag torque(kg-mm)
Motor drive torque during acceleration(kg-mm)
Output torque(N-mm)
Total operating torque(kg-mm)
Maximum drive torque(kg-mm)
Ballscrewweight(kg)
Total weight(mm)
Number of balls
Greek Letters:
µ
α
αm
φ
𝜂1
𝜂2
δ1,2
σ1,2
f
µk
Friction coefficient (0.005~0.02)
Initial contact angle of support bearing
Maximum deformation(mm)
Lead angl(degrees)
Mechanical efficiency(0.85~0.95)
Mechanical efficiency(0.75~0.85)
Geometric constant(mm2/N)
Principal stresses(MPa)
Rolling friction coefficient(0.003~0.01)
Friction coefficient(0.029~0.12)