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Advances in Technology Innovation, vol. 2, no. 2, 2017, pp. 29 - 33 

29 

Investigation of a Ball Screw Feed Drive System Based on 

Dynamic Modeling for Motion Control 

Yi-Cheng Huang*, Xiang-Yuan Chen 

Department of Mechatronics Engineering, National Changhua University of Education,  

Changhua, Taiwan. 

Received 01 February 2016; received in revised form 28 April 2016; accepted 02 May 2016 

 

Abstract 

This paper examines the frequency response 

relationship between the ball screw nut preload, 

ball screw torsional stiffness variations and table 

mass effect for a single-axis feed drive system. 

Identificat ion for the frequency response of an 

industrial ball screw drive system is very im-

portant for the precision motion when the vi-

bration modes of the system are critical for 

controller design. In this study, there is transla-

tion and rotation modes of a ball screw feed  

drive system when positioning table is actuated 

by a servo motor. A lumped dynamic model to  

study the ball nut preload variation and torsional 

stiffness of the ball screw drive system is de-

rived first. The mathematical modeling and  

numerical simulat ion provide the informat ion of 

peak frequency response as the different levels  

of ball nut preload, ball screw torsional stiffness 

and table mass. The trend of increasing preload 

will indicate the abrupt peak change in fre-

quency response spectrum analysis in some 

mode shapes. This study provides an approach 

to investigate the dynamic frequency response of 

a ball screw drive system, which provides  sig-

nificant information fo r better control perfor-

mance when precise motion control is con-

cerned. 

Keywords: Ball nut preload, ball screw drive 

system, dynamic modeling 

1. Introduction 

Precision computer numerical control (CNC) 

machines are widely used in  modern  industry for 

mass production. The ball screws are widely  

applied in the linear actuators of machinery and  

equipment because of the high efficiency, less 

backlash, easy lubrication, and easy maintenance. 

Since ball screw play a significant role in con-

verting rotary motion into linear mot ion Pre-

loading is effective to eliminate backlash and 

increase the stiffness of ball screw for precision  

motion concerns. The dynamic frequency re-

sponses of the feed drive system depends on the 

stiffness combinations of the ball screw, ball nut, 

fixed support bearings, the flexible coupler and  

the stiffness between the ball screw and the 

working table. Such frequency response results 

from the axial mode shapes and torsional mode 

shape of the ball screw drive system when it is  

actuated by servo motor. Each mode shape af-

fects and determines the motion control fre-

quency response bandwidth when the control 

speed is limited and becomes a critical issue. 

The working table mass and the bolt stiffness 

between the machine bed base and attached 

ground floor also plays an important role for the 

frequency response when the vibration mode of 

the CNC machine is concerned with precision  

accuracy. 

As in intelligent control field, the control 

signals that fed into the controlled plant are 

based on the feedback control error that should 

be learnable. Therefore, some control efforts 

[1-2] are focused the design of the bandwidth of 

the filter that can filter out the un-learnable er-

rors contents and can be get back to control 

system for bettering control h istory. Since the 

bandwidth of controlled system determines the 

motion speed response and its performance. The 

lumped dynamic model derivation  is significant  

in determin ing the bandwidth for motion control 

law that can be used for controller design. Since 

such frequency contents of compensated error 

are suggested to be within the bandwidth when 

control signals are actuating. Th is paper will 

derive the dynamic model of the ball screw feed  

drive system first and examine the relationship 

between the ball screw preload variation, ball 

screw torsional stiffness  and effect of table mass. 

* Corresponding author, Email: ychuang@cc.ncue.edu.tw 



Advances in Technology Innovation, vol. 2, no. 2, 2017, pp. 29 - 33 

30 Copyright ©  TAETI 

Simulation results will unveil the frequency 

response of the translational and torsional peak 

modes. Numerical simulat ion shows stable 

convergence by using hybrid particle swarm 

optimization of iterative learning control [2] on  

this developed ball screw drive system. 

2. Mathematical Model and Numer-

ical Simulation 

2.1. Dynamic Model of the Ball Screw Drive 

System 

 To model the feed drive system, th is paper 

set differnt stiffness of the ball nut stiffness for 

different preload between the ball screw shaft 

and the ball nut. The presetting preload value 

can be deployed by inserting different ball size 

for single ball nut design or using disk spring 

that applied to the ball screw when double ball 

nut is the preference. Fig. 1 shows the picture of 

the in-lab single-axis feed drive p latform. To  

analyze the dynamic characteristic of the ball 

screw system under different preload and vary-

ing table mass, the feed drive system is modeled  

by a lumped parameter system shown in Fig. 2. 

Fig. 2 is the schematic illustration for the sin-

gle-axis ball screw feed drive system. In general, 

mechanical systems have three passive linear 

components. The spring and the mass are ener-

gy-storage elements, while the viscous damper 

is the dissipated energy. Both of the rotational 

and translation mechanical system modeled  

below are actuated by the servo motor torque, 

indicated as T. As the same derivation in [3], the 

overall stiffness of a ball screw feed drive sys-

tem can be determined by the stiffness of the ball 

screw itself, which is comprised of the ball 

screw shaft, the ball nut, supporting bearings of 

the ball screw, and the stiffness between the ball 

screw and the working table. 

 )( T 
bm

gKmmQmmJ







 

 

(1) 

𝐽𝑏 × �̈�𝑚 + 𝑄𝑚 × �̇�𝑏 + 𝑅 ×
[𝐾𝑛 × (𝑅 × 𝜃𝑏 + 𝑋𝑏 − 𝑋𝑡

)] =
𝐾𝑔 × (𝜃𝑚 − 𝜃𝑏

)  

(2) 

𝑀𝑏 × �̈�𝑏 + 𝐵𝑏 × (�̇�𝑏 − 0) +

𝐾𝑒 × (𝑋𝑏 − 0) + 𝐾𝑛 ×
(𝑅 × 𝜃𝑏 + 𝑋𝑏 − 𝑋𝑡

) = 0  

(3) 

 tX
b

X
b

RnK

tXtBtXtM

)(

)0(









 

(4) 

Rearranging Eqs (1)-(4), we have 

[

𝑀𝑡 0 0 0
0 𝑀𝑏 0 0
0 0 𝐽𝑏 0
0 0 0 𝐽𝑚

] ×

[
 
 
 
 
�̈�𝑡

�̈�𝑏

�̈�𝑏

�̈�𝑚]
 
 
 
 

+

[

𝐵𝑡 0 0 0
0 𝐵𝑏 0 0
0 0 𝑄𝑏 0
0 0 0 𝑄𝑚

] ×

[
 
 
 
 
�̇�𝑡

�̇�𝑏

�̇�𝑏

�̇�𝑚]
 
 
 
 

+

[
 
 
 

𝐾𝑛 −𝐾𝑛 −𝐾𝑛𝑅 0

−𝐾𝑛 𝐾𝑒 + 𝐾𝑛 𝐾𝑛𝑅 0

−𝐾𝑛𝑅 𝐾𝑛𝑅 𝐾𝑔 +𝐾𝑛𝑅2 −𝐾𝑔

0 0 −𝐾𝑔 𝐾𝑔 ]
 
 
 

×

[

𝑋𝑡
𝑋𝑏
𝜃𝑏

𝜃𝑚

] = [

0
0
0
𝑇

]  

(5) 

where the stiffness matrix is different from [3]. 

 

Fig. 1 The in-house single axis platform 

 

Fig. 2 Illustration for the schematic d iagram of 

the single-axis lumped  parameters ball 

screw drive system 



Advances in Technology Innovation, vol. 2, no. 2, 2017, pp. 29 - 33 

31 Copyright ©  TAETI 

2.2. Numerical Simulation 

Table 1 list the simulated parameters and 

associated values used in Eqs (1)-(4). The dy-

namic equation of the single-axis feed drive 

system model with varied p reload and table 

mass can be expressed in a compact form: 

[𝑀]{�̈�} + [𝐶]{�̇�} + [𝐾]{𝑢} = 𝑓  (6) 

where [M], [C], and [K] are the 4x4 square ma-

trices, referred as the mass (or the moment of 

inertia), the viscous damping, and the stiffness 

matrices, respectively. {u} represents a four 

degree of freedom model. It consists with the Xt , 

Xb , θb , and θm  for the displacement of the 

working table, axial displacement of the ball 

screw, rotation angle of the ball screw, and the 

rotation angle of the motor, respectively.  

Table 1 Important parameters of ball screw drive 

system 

Parameters value 

working table mass (Mt) 47.09Kg 

ball screw mass (Mb) 9Kg 

inertia moment of the motor (Jm) 4.45× 10−4kgm2 

inertia moment of the ball screw 

(Jb) 
1.3× 10−3kgm2 

equivalent axial stiffness of ball 
screw shaft (Ke) 

1.8663× 107N/m 

stiffness of the ball nut (Kn) 2.3345× 108N/m 

torsional stiffness of the ball 
screw  (Kg) 

3.49× 108N/m 

viscous damping coefficient of 

the guide way of the working 
table (Bt) 

10N s/m 

viscous damping coefficient of 

the supporting bearing of the ball 

screw (Bb) 

10N s/m 

rotational viscous damping co-

efficient of the motor (Qm) 
0N ms 

rotational viscous damping co-

efficient of the support bearing of 

ball screw (Qb) 

0N ms 

angle conversion axial dis-

placement of the constant (R) 
0.0025 

motor torque (T) 1.8× 10−3Nm/𝑠2 

displacement of the working 

table (Xt) 
State Variable (m) 

axial displacement of the ball 

screw (Xb) 
State Variable (m) 

rotation angle of the motor (θm) State Variable (rad) 

rotation angle of the ball screw 

(θb) 
State Variable (rad) 

The homogeneous solution of Eqs (5) rep-

resents the transient response of the lumped  

system whereas the forcing function of applied  

motor torque renders the table positioning. 

Homogeneous solution of the Equation (2.1.5) 

results in four eigenvectors V1, V2, V3, and V4 

associated with each eigenvalue ( λ = ω2 ) of  

3.2124 × 104 ,  3.3322 × 106  , 1.0699 ×
1011  ,3.421 × 107(rad2/s2).  

V1={

−0.7299
−0.6836

0
0

}V2={

0.1762
−0.9844

0
0

} 

V3 ={

0
0

0.9482
−0.3176

}V4={

0
0

0.7067
0.7076

} 

(7) 

The four eigenvectors corresponding to the 

Eigen frequencies of 28.52 Hz, 290.52 Hz, 

52059 Hz, 930.9 Hz w is calculated by 2% of the 

rated dynamic load. These eigenvectors repre-

sent the mode shapes of the ball screw feed drive 

system. The first three significant Eigen fre-

quencies are related to the three resonant fre-

quencies. The first and second modes are from 

the axial vibration. The third mode is from the 

torsional vibration. In Eqs (7), the first mode of 

the working table and the ball screw is moving in  

phase while the second mode is moving out of 

phase. Fig. 3 shows the bode plot of the dynamic 

system based on different Kn values. The pre-

load variation is simulated from 2%, 4%, 6%, 

8% to 10% of the rated dynamic loading of the 

ball screw. 

 

Fig. 3 Bode plot of the ball screw drive system 

based on the preload value of 2%(blue),  

4%(green), 6%(red), 8%(cyan) and 

10%(purple) of the rated dynamic loading 



Advances in Technology Innovation, vol. 2, no. 2, 2017, pp. 29 - 33 

32 Copyright ©  TAETI 

Enlargement of the first, second and third 

modes of the bode plot, the three frequencies are 

ranging from 30.5 Hz to 30.8 Hz, 294 Hz to 380 

Hz and 52000 Hz to 55000 Hz respectively. It is  

obvious that the second translational mode will 

be affected more than the first mode when the 

ball nut preload is vary ing. The solution of the 

fourth mode is about 930 Hz, the contribution of 

this mode is not significant even though the 

preload is varied. 

 

Fig. 4 The enlargement of the first mode of the 

bode plot in Fig. 3 based on different ball 

nut stiffness 

 

Fig. 5 The enlargement o f the second mode of 

the bode plot in Fig. 3 based on different 

ball but stiffness 

 

Fig. 6 The en largement of the third mode of 

bode plot in Fig. 3 based on different ball  

nut stiffness 

As stated, the third eigenvector indicates the 

torsional vibration. Since the servo motor drives 

the ball screw through a coupler providing  

damping and stiffness. The torsional stiffness of 

the ball screw is investigated by calculating  

Kg   by  πdr
4G/32L  . Fig. 7 shows the variations 

of  Kg in the range of ±  10%. As shown, the 

third mode demonstrates large frequency shift 

with increasing the torsional stiffness of the ball 

screw. Fig. 8 indicates the frequency shift is  

range from  50700 Hz to 58000 Hz, while the 

first and second modes are not changed notice-

ably. 

 

Fig. 7 Bode plot of the ball screw torsional 

stiffness 100%( blue), 105%(green), 

110%(red), 95%(cyan), 90%( purple) 

 

Fig. 8 The enlargement of the torsional  mode of 

the bode plot in Fig. 7 based on  different 

ball screw torsional stiffness 

Figs. 9-11 detail the effect of the table mass. 

The characteristic frequency shifts from the 

24Hz to 30.5Hz when the table mass increases 

from 47.09 kg to 94.18 kg. As predicted, the 

table mass preserves the effect on the first 

bandwidth in the translational mode and some 

effect on the second mode. Increasing table mass 

does not affect the bandwidth of the third tor-

sional mode. Fig. 12 shows the numerical sim-

ulation plot of the convergence error when the 

ball screw drive system is controlled by hybrid  

particle swarm optimization for the dynamic 

bandwidth tuning of an iterative learning con-

trol. 



Advances in Technology Innovation, vol. 2, no. 2, 2017, pp. 29 - 33 

33 Copyright ©  TAETI 

 

Fig. 9 Bode plot of the working table mass 

47.09Kg(blue), 70.63Kg(green), 

94.18Kg(red) 

 

Fig. 10 The en largement of the first mode of 

the bode plot in Fig. 9 based on  dif-

ferent working table mass 

 

Fig. 11  The enlargement of the second mode 

of the bode plot in Fig. 9 based on 

different working table mass 

 

Fig. 12 Plot of a convergence error of the nu-

merical s imulat ion by  using  

HPSO-ILC [2] for ball screw drive 

system 

3. Conclusions 

A lumped dynamic model fo r describing  

different ball nut preload level, ball screw tor-

sional stiffnesss and the table mass effects of the 

ball screw feed drive system is derived and 

numerically simulated. Based on the different  

percent of the preload and table mass, the fre-

quency spectrum analysis of the numerical sim-

ulation provides the limits and constraints of 

bandwidth tuning for motion control applications. 

The preload variation can be diagnosed by the 

peak frequency change and the magnitude of the 

peak frequency in a specific frequency range 

when the axial mode or the torsional mode is  

excited. The derivation and numerical simulation  

results of the lumped dynamic model provides 

significant informat ion in determining the zero  

phase bandwidth tuning. Application of a hybrid  

particle swarm optimization iterat ive learning  

control law deploys successfully when the fre-

quency contents  of the compensated error was 

constrained in every control actuation. 

Acknowledgement 

This work was supported by MOST Grant  

104-2221-E-018-015 for which the authors are 

very much grateful. 

References 

[1] M. S. Tsai, C. L. Yen, and H. T. Yau, “In-

tegration of an empirical mode decomposi-

tion algorithm with iterative learn ing con-

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IEEE/ASME Transaction on Mechatronic, 

vol. 18, no. 3, pp. 878-886, 2013. 

[2] Y. C. Huang, Y. W. Su, and P. C. Chuo, 

“Iterative learn ing control bandwidth tuning 

using  the part icle s warm opt imizat ion  

techn ique fo r h igh p recis ion mot ion,”  

Microsystem Technologies. DOI: 10.1007/

s00542-015-2649-6, 2015. 

[3] G. H. Feng and Y. L. Pan, “Investigation of 

ball screw preload variation based on dy-

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