Analytical Solution of the Frequency

(1)

Design and Dynamics

Analytical Solution of the Frequency

Expressions for Rigid-Body Natural Vibration of a Carriage on Linear Guideway Type

Recirculating Rollers*

Thin-Lin HORNG**

**Department of Mechanical Engineering, Kun-Shan University, Tainan, Taiwan, R.O.C.

Email : hortl@mail.ksu.edu.tw

Abstract

The analytical solution of vertical, pitching, yawing, lower rolling, and higher rolling frequency expressions for linear guideway type (LGT) recirculating rollers with arbitrarily crowned profiles was explored. Each roller inside LGT recirculating roller is divided into three parts: two crowned and one cylindrical. The superposition method is introduced to obtain the stiffness equation for rollers compressed between the carriage and profile rail, and a model of discrete normal springs is used to construct the stiffness equation for LGT recirculating roller. Five equations of motion were obtained using Lagrange's equation. The results reveal that the frequencies are affected by the value of preload and of stiffness. The natural frequency of the exponential profile is close to that of the forth power profile. Therefore, recirculating rollers with an exponential profile, a small crowned depth, and a large crowned length seem to be optimal, having a higher frequency of LGT recirculating roller and a lower edge stress concentration.

Key words: Linear Guideway Type Recirculating Rollers, Natural Frequency Expressions, Stiffness Equation, Roller, Arbitrarily Crowned Profiles

1. Introduction

The linear motion roller guideway shown in Fig.1 has many advantages compared to ball guideways [1] and conventional sliding guides, such as flat ways and V-ways [2]. For instance, the ultimate loading of the roller guideway can be larger than that of a ball guideway, making the lubrication more efficient so that the abrasion of linear motion roller bearings is less than that of sliding guides. Linear motion roller bearings also have no stick slip, and are widely used instead of ball guideways in heavy CNC machining centers, grinding machines, precision heavy-duty X-Y tables, and TFT-LCD transport systems [3].

In recent years, the speed of machines using linear motion rolling bearings has increased. As a result, the sound and vibration of linear motion rolling bearings have contributed to the serious problem of noise and vibrations of these machines [4-5]. Kasai et al. [6-7] found that carriages of preloaded recirculating linear roller bearings moved periodically with the roller passing frequency due to roller circulation. Ye et al. [8] carried out a modal analysis for carriages of linear guideway type (LGT) recirculating linear roller bearings under stationary conditions and pointed out the existence of the rigid-body natural vibrations of the carriage. Schneider [9] developed a theory for the natural vibrations of an LGT recirculating linear ball bearing with a 45 degree contact angle. The analytical analyses of frequency expressions for a linear motion roller guideway with recirculating rollers, which are a combination of straight and curved profiles, were not developed in the mentioned studies.

The vibration frequency of the roller/carriage and roller/rail are often used to

*Received 6 Oct., 2008 (No. 08-0705) [DOI: 10.1299/jsdd.3.215]

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Dynamics

determine the operational parameters and the dynamic behavior of the linear motion roller guideway. Thus, the frequency analyses of LGT recirculating roller are important and necessary. Generally, this problem can be solved using the finite element method; however, FEM requires a complicated mesh and a time-consuming contact analysis. An alternative is to develop an analytical method that evaluates the LGT recirculating roller vibration frequencies efficiently and easily. It is important to reduce edge stresses on the roller contact surface and to obtain a substantially uniform contact stress distribution. This has been attained primarily by using specially crowned profiles for the contact surface [10].

Nayak [11] presented a method of calculating the pressure between elastic bodies having a slender area of contact and arbitrarily profiles. He used Lundberg’s equation [12] in his study and suggested that the combined compressive deformation of both bodies (the stiffness between elastic bodies) should be used to construct a compatible deformation equation. Horng et al. [13] demonstrated a stiffness equation for the circularly crowned roller compressed between two plates. In order to obtain the optimum roller contact surface, the stiffness equation was modified to describe the stiffness other than the circularly crowned roller [14-15].

In the present study, an analytical solution of the frequency expressions for rigid-body natural vibration of a carriage on linear guideway type recirculating rollers with arbitrarily crowned profiles is explored. Recirculating rollers compressed between a carriage and a profile rail are simulated as rollers compressed between two plates, and the superposition method is introduced to obtain the stiffness equation for a crowned roller compressed between a carriage and a profile rail; rollers with circular, quadratic, cubic, forth-order-power, and exponential profiles were analyzed. Then, a model of discrete normal springs is used to obtain the stiffness equation for LGT recirculating rollers with arbitrarily crowned profiles. Finally, five equations of motion were obtained using Lagrange's equation, and the analytical solution of the frequency expressions was explored.

These frequency expressions include the vertical, pitching, yawing, lower rolling and higher rolling expressions. The results reveal that recirculating rollers with an exponential profile, a small crowned depth, and a large crowned length seem to be optimal for obtaining higher LGT recirculating roller stiffness and a lower edge stress concentration.

2. Stiffness Formula for LGT

Figure 1 shows the configuration of an LGT recirculating roller bearing. In order to analyze the stiffness equation, the stiffness formula for a crowned roller compressed by a carriage and a profile rail is investigated. The model and dimensions are shown in Fig 2 and the profiles of a crowned roller are shown in Fig 3. The crowned roller is divided into three parts, two crowned parts and one cylindrical part. The primary axes of the roller system, denoted by

x

and

y

, are shown in Fig.3. The assumptions of this stiffness formula are: (1) friction is neglected because of the lubrication of the roller, (2) the depth effect of the carriage and rail can be simulated as an equivalent depth and (3) the solution is obtained for the small-strain and linear elastic conditions. In the following sections, we derive this stiffness formula.

Oil inlet Metal wiper

Carriage Side cover

Profile rail

Recirculating module Crowned

Roller Holder

a Applied load

Screw cover

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Dynamics

x

r

C

Fig.2 Model and dimensions of a crowned roller compressed between a carriage and profile rail, or one discrete spring.

Fig.3 Geometry of arbitrarily crowned profiles.

Using the superposition method, the stiffness formula for a crowned roller compressed by a carriage and profile rail,

K

, can be written as[14]:

δ

δ ⁿ ^c ⁿ ^c

P K P

Q K

K= = +2 = +2 (1)

where

K

_n and _K_c are the stiffness contribution of the non-crowned part and that of the crowned part, respectively; Qis the total displacement applied on a roller; and δ, P_n^{, and}

P

c are the total displacement, the load applied on the non-crowned part, and that applied

ds

t

s dt

L

Arbitrarily crowned part

Non-crowned part Arbitrarily crowned part

H Deformation

H

y

z

2c

Profile rail Carriage

Crowned roller (Discrete spring)

Displacements applied on this plane

x

y

r

C

y

z

(4)

Dynamics

on the crowned part, respectively. They can be calculated by[14]:

[ ]















 



 + −

 −



 



 



 



 + −

 −



 



=  ⁻ ⁺

− −

2 / 21 1)(1( / ) 1

( 2

1

2 2 2 2 1 1 1

2 2 2 2 1 1

1 ln 1

1 1

4 _H _c

n

n E E e

P H c E

E c

P _ν

ν π ν

δ (2)

where

ν

₁ is Poisson's ratio of the carriage and profile rail ,

E

₁ is Young's modulus of the carriage and profile rail,

ν

₂ is Poisson's ratio of the roller,

E

₂ is Young's modulus of the roller,

2 c

is the length of the non-crowned part and

H

is the equivalent thickness of the carriage and profile rail.

2 / 3 1 1

2 2 2 2 1 1

9 2 1

2 1











 



 





′ ℑ



 



 + −

= −

−

y c x c

E R E

P E ε δ

ν

πκ ν (3)

where

R

_x_y is the curvature sum,

δ

_c is the deformation between the profile rail and crowned part of the roller,

κ

is the ellipticity parameter,

ℑ

is the elliptic integrals of the first kind and,

ε

is there of the second kind.

The stiffness shown in Eq.(1) is not an explicit form of the total applied force; thus, the total displacement δ is defined first. Eq. (2) and (3) can be used to obtain

P

_nand

P

_c. Finally, Eq.(1) can be used to evaluate the roller stiffness. In other words, one crowned roller is modeled as a spring, with the spring constant calculated using Eq.(1).

In this paper, the crowned profiles shown in Fig.5 were investigated. Except for the exponential profile, the intersection point of the straight line and the profile function is set to be the original point, as shown in Fig.4.

(1) Circular profile (Fig.4a): the profile equation is:

2 2

s r r

t

= _C_y− _C_y − (4) where

r

_C_y= (ds²+dt²)/ (2dt). In this paper, dt<<ds, t≅s²/(2

r

_C_y), so t≅ (dt/ds²) s² , which is the same as the quadratic profile. In other words, the quadratic profiles can be approximated by the circular profile when dt<<ds.

(2) Cubic profile (Fig.4a): the profile equation is t = (dt/ds³) s³. (5) (3) Fourth-order power profile (Fig.5a): the profile equation is t = (dt/ds⁴) s⁴. (6) (4) Exponential profile (Fig.4b).

Since the exponential function is always greater than zero, the intersection point of the straight line and the exponential function is assumed to be at s=0 and t=dt/n, where n is a parameter. In this study, n is set to 100. The profile equation is:

t=aExp(bs), where

a = dt / n

and

ds

1) + Log(n

=

b (7)

In the mathematical sense, the logarithm function is the inverse function of the exponential function, so the two functions are symmetric along the line Y=X; thus, the smooth behavior of the two functions is the same. We do not include the logarithm function in this paper. Except for the linear and exponential profiles, the tangent directions of the straight line and the crowned profile at the intersection point are equal to zero.

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Dynamics

∆

∆ s s t t

Quadratic t=as Cubic t=as Fourth-power t=as

2 3

4 Circular

∆

∆s

t s t

∆t/n

t=a*Exp(bs) Exponential

(a) (b)

Fig.4 Arbitrarily crowned roller profiles.

Rollers compressed between a carriage and profile rail are shown in Fig. 1 and configuration and dimensions of a LGT recirculating roller bearing are demonstrated as Fig.

5. Fig.6 show a section AA denoted in Fig.5 along the longitude direction of a LGT recirculating rollers, which are simulated as springs connected in parallel and caused by the elastic contact of carriage and profile rail. Each discrete normal spring was modeled as a crowned roller compressed between two plates, as shown in Fig.2. In other words, a roller compressed between two plates shown in Figs. 2, are used to simulate a roller compressed between carriage and profile shown in Figs. 6. The depth effect of a carriage and profile rail is simulated as the depth of plates. In an LGT recirculating linear roller bearing without an external force, the normal force is applied to each contact point of the raceways and the rollers in the load zone, with normal elastic deformation at each contact point. Thus, each contact point has the characteristics of a spring. Since the springs exist at interval

s

of the loaded rollers, they are named "discrete normal springs". The carriage is supported by the discrete normal springs in the load zone of each circuit of the recirculating rollers. The discrete normal stiffness is denoted by

K

, as shown in Fig. 6. When the LGT recirculating linear roller bearing is driven at a constant velocity, each contact point of the raceways and rollers constantly changes. Moreover, the total number of rollers in the load zone varies. As a result, the location and number of the discrete normal springs change, and stiffness

K

varies. These changes should be considered in the theoretical analysis of the rigid-body natural vibration of the carriage. However, it is very difficult to consider these changes from a theoretical point of view. In this paper, the discrete normal springs are replaced by distributed normal springs, which are continuously distributed along the length of the load zone of each roller circulation.

Fig.5 Configuration and dimensions of a LGT recirculating roller bearing.

dt/n

ds

dt

y z

ψ

u θ v φ

α

c

A

A b

a

o

ds

dt

(6)

Dynamics

The normal stiffness equation for LGT recirculating rollers with arbitrarily crowned profiles, denoted by

K

_V, can be obtained as [5]:

K Z

K

_V

= ( 4

_L

sin

²

α )

(8) where

Z

_Lis the average number of rollers in the load zone in one circuit of the recirculating rollers,

α

is the contact angle shown in Fig. 5, and

K

is the discrete normal spring constant shown in Fig. 6.

Fig.6 Configuration of Section AA denoted in Fig.5 along the longitude direction of LGT and springs caused by the elastic contact of raceways and rollers.

3. Frequency Expressions for Rigid-Body Natural Vibration of a Carriage for LGT

In Figure 5, the origin

o

of the coordinates

xyz

coincides with the position of the center of gravity of the carriage when it is not vibrating. Although the

x

-axis is not shown in Figure 8, it is parallel to the longitudinal direction of the profile rail. Since the

x

-axis is also parallel to the driven direction of the carriage, the displacement of the carriage along the

x

-axis is not considered.

α

is the contact angle shown in Fig.5;

a

is the distance from the origin (

o

) to the contact point of the upper circuits of recirculating rollers in the carriage and distributed normal springs in the direction parallel to the

z

-axis,

b

o

) to the contact point of the lower circuits of recirculating rollers in the carriage and distributed normal springs in the direction parallel to the

z

-axis;

and

c

o

) to the contact point of the carriage and distributed normal springs in the direction parallel to the

y

-axis. Using Eq.8, the spring constant

k

per unit length of the distributed normal springs can be written as:

α sin

2

4

_L

V L

L

Z K l

K

k = Z =

(9)

where

l

_Lis the length of the load zone in one circuit of the recirculating rollers shown in Fig. 6.

When the carriage supported by the distributed normal springs vibrates in rigid body mode, Lagrange's equation can be used to obtain the following five equations of motion:

K

) ( l

_L

Section AA

z x

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Dynamics

2 2

(4

_L

cos ) {2

_L

( ) cos } 0

Mu + kl α u − kl a b + α φ =

(10)

(4

_L

sin

2

) 0

M ν + kl α ν =

(11)

2 2 2 2

2 2

{2 ( ) cos } {2 ( ) cos

4 ( ) sin cos 4 sin } 0

x L L

L L

J kl a b u kl a b

kl a b c kl c

φ α α

α α α φ

− + + +

+ − + =

(12)

3 2

( sin ) 0

y

3

L

J θ + k l α θ =

(13)

3 2

( cos ) 0

z

3

L

J ψ + k l α ψ =

(14)

where

M

is the mass of the carriage;

J

_x,

J

_y and

J

_zare the moments of inertia about the

x

,

y

, and

z

axes, respectively;

φ

,

θ

and

ψ

are the angular displacement of the carriage around the

x

,

y

, and

z

axes, respectively;

u

and

v

are the displacements of the carriage in the direction of

y

and

z

axes, respectively.

For convenience, the natural vibration with the translational motion mode along the z-axis is called the vertical natural vibration of the carriage, the natural vibration with the rotary motion mode around the y-axis is called the pitching natural vibration of the carriage, and the natural vibration with the rotary motion mode around the z-axis is called the yawing natural vibration of the carriage. Therefore, the frequency

f

_vof the vertical natural vibration of the carriage, the frequency

f

_pof the pitching natural vibration of the carriage, and the frequency

f

_Yof the yawing natural vibration of the carriage are given by:

M f

_v

kl

^L

π α

= sin

(15)

y L

p

J

f kl

3 2

sin

³

π

= α

(16)

z L

Y

J

f kl

3 2

cos

³

π

= α

(17)

The frequency equation can be expressed as:

0 )

(

₃ ₁ ² ₁ ₃ ₂²

4

− c M + c J + c c − c =

MJ

_x

ω

_x

ω

(18)

[ ] ^ ^ _

 





+

− +

+

=

+

−

=

α α

α

α α α

2 2

2 2 2 3

2 2

2 1

sin 2 cos sin ) ( 2

cos ) (

2 cos ) ( 2

cos 4

c c

b a

b a kl c

kl c

L L

L

(19)

The two solutions

ω

₁²and

ω

₂²,(

ω

₁²<

ω

₂²), are given by:

x

x x

x

MJ

c c c MJ J

c M c J c M c

2 ) (

4 )

(

₃ ₁ ² ₁ ₃ ₂²

1 3 2

2 , 1

−

− +

= +

−

ω

+ (20)

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Dynamics

In Eq(20), the negative sign of the radical sign is for

ω

₁²and the positive sign is for

2

ω

2 . In this paper, the coupled vibration with lower frequency is called the lower rolling natural vibration of the carriage

f

_RL, and the coupled vibration with higher frequency is called the higher rolling natural vibration of the carriage

f

_RH. From Eq. (20), the frequency

f

RL of the lower rolling natural vibration of the carriage and the frequency

f

_RH of the higher rolling natural vibration of the carriage are given by:

π ω 2

=

1

f

RL (21)

π ω 2

=

2

f

RH (22)

4. Analysis Using Frequency Expressions for Rigid-Body Natural Vibration of a Carriage

Using the above frequency expressions, we calculated the natural frequencies of the rigid-body natural vibrations of a carriage on LGT recirculating rollers. In the calculation of the natural frequencies, the dimensions and material properties of LGT recirculating rollers shown in Table 1, and the values of

M

,

J

_x,

J

_y,

J

_z,

a

,

b

, and

c

listed in Table 2 were used. Linear guideway type recirculating rollers with arbitrarily crowned profiles are shown in Fig.1. Due to symmetry, rollers compressed between a carriage and profile rail are simulated as springs connected in parallel, as shown in Fig. 6. Each discrete normal spring was modeled as a crowned roller compressed between two plates. Figure 2 shows this model and its dimensions. The boundary conditions of the model in Fig.2 and Fig.6 are (1)

y

-direction of springs representing the rollers along the leftmost surface, (2)

x

-direction of springs representing the rollers along the back surface, (3) the same

z

-direction displacement applied on the top surface, (4)

z

-direction rollers along the bottom surface, (5) contact elements between the spring and carriage or profile rail, and (6) free for other boundaries.

A total displacement of 0.005 mm, divided into ten displacement increments, was uniformly applied to the upper surface of LGT recirculating rollers.

Table 1 Test linear bearing specifications

Total length of a roller, 2c 6 mm

Effective depth of the carriage/profile rail,

H

^{20 mm}

Radii of roller,

r

_C_x ^{2 mm}

Crowned length of roller,

ds

^{0.8 mm}

Crowned depth of roller,

dt

^{0.008 mm}

Young's modulus of the carriage/profile rail,

E

₁ ^206Gpa

Young's modulus of the roller,

E

₂ ^314Gpa

Poisson's ratio of the carriage/profile rail,

ν

₁ ^0.29

Poisson's ratio of the roller,

ν

₂ ^0.3

The average number of rollers in the load zone in one circuit of the recirculating rollers,

Z

_L

22

Number of circuits of recirculating rollers 4

Contact angle,

α

50 degree

Load zone length,

l

_L ^{108 mm}

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Dynamics

Table2. Values of

M

,

J

_x,

J

_y,

J

_z,

a

,

b

, and

c

The mass of the carriage,

M

^{0.465 kg}

The moments of inertia about the

x

axis,

J

_x ¹⁴³^.⁹^kg−^mm²

y

axis,

J

_y ¹⁹³^.⁷^kg⁻^mm²

z

axis,

J

_z ²⁵⁷^.⁶^kg−^mm²

The distance from the origin (

o

) to the contact point of the upper circuits of recirculating rollers in the carriage and distributed normal springs in the direction parallel to the

z

-axis,

a

mm 18 . 4

o

) to the contact point of the lower circuits of recirculating rollers in the carriage and distributed normal springs in the direction parallel to the

z

-axis,

b

mm 8 . 13

o

) to the contact point of the carriage and distributed normal springs in the direction parallel to the

y

-axis,

c

mm 0 . 13

5. Results and Comparisons

Stiffness analyses were performed for each crowned profile. Figure 7 shows a comparison of stiffness for LGT recirculating rollers with arbitrarily crowned profiles.

When dt<<ds, the circular profile is very similar to the quadratic profile. Figure 7 indicates that: (1) the stiffness of LGT recirculating rollers increases when the applied load increases;

(2) a higher-order power profile has higher stiffness; (3) the stiffness of the exponential profile is close to that of the forth power profile. This is because a higher-order power has a smooth crowned curve, resulting in higher edge stress. Therefore, an exponential profile seems to be an optimal solution for LGT recirculating rollers.

Fig.8 shows a comparison of calculated natural frequencies of a carriage for LGT recirculating rollers with exponential crowned profiles. Figure 8 indicates that the stiffness of LGT recirculating rollers significantly affects the natural frequencies; all five frequencies significantly increase when the applied displacement increases. This is because the stiffness became large under a large applied displacement. Fig.9 and Fig.10 show a comparison of the vertical natural frequency and low rolling natural frequency of a carriage on LGT recirculating rollers with exponential crowned profiles, respectively. Fig. 9 and Fig.10 show that the vertical and low rolling natural frequencies increase when the applied load increases. An exponential profile seems to have the largest natural frequency. This is also true for the other natural frequencies.

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Dynamics

0 1 2 3 4 5

Applied Displacement (1/1000 mm) 3.5E+006

4.0E+006 4.5E+006 5.0E+006 5.5E+006 6.0E+006

Stiffness (N/mm)

Stiffness of LGT with arbitrarily crowned roller

Circular profile Quadratic power profile Cubic power profile Forth power profile Exponential profile

Fig.7 Comparison of stiffness for LGT recirculating rollers with arbitrarily crowned profiles.

0 1 2 3 4 5

Applied Displacement (1/1000 mm)) 100

200 300 400 500 600

Frequency (Hz)

Natural Frequency of a Carriage Vertical

Pitching Yawing Low Rolling High Rolling

Fig.8 Calculated natural frequencies of a carriage on LGT recirculating rollers with exponential crowned profiles.

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Dynamics

0 1 2 3 4 5

Applied displacement (1/1000 mm) 400

440 480 520 560 600

Frequency (Hz)

The Vertical Frequency Circular Profile Quadratic Profile Cublic-Power Profile Forth-Power Profile Exponential Profile

Fig.9 Comparison of the vertical natural frequencies of a carriage on LGT recirculating rollers with arbitrarily crowned profiles.

0 1 2 3 4 5

Applied Displacement (1/1000 mm) 180

200 220 240 260

Frequency (Hz)

The Low Rolling Frequency Circular profile Quadratic Profile Cubic-Power Profile Forth-Power Profile Exponential profile

Fig.10 Comparison of the low rolling natural frequencies of a carriage on LGT recirculating rollers with arbitrarily crowned profiles.

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Dynamics

6. Conclusion

In this paper, linear guideway type (LGT) recirculating rollers were modeled as discrete normal springs. Then, an analytical solution of the frequency expressions for rigid-body natural vibration of a carriage on linear guideway type recirculating rollers was developed.

The crowned roller was separated into three parts: one cylindrical and two crowned. A model of discrete normal springs was used to construct the stiffness equation for LGT recirculating rollers. The analytical solution of vertical, pitching, yawing, lower rolling, and higher rolling frequency expressions was explored using Lagrange's equation. The simulation results reveal that a higher-order power can make an LGT recirculating rollers with higher frequencies. The frequency of the exponential profile is close to that of the forth power profile. Therefore, recirculating rollers with an exponential profile, a small crowned depth, and a large crowned length seem to be optimal for obtaining a higher natural frequency of LGT recirculating rollers and lower edge stress concentration.

Acknowledgment

This study was support by the National Science Council, Republic of China, under contract number: NSC 97-2622-E-168-006-CC3.

References

(1) HIWIN Co., Ltd., Technical Information of Linear Guideway, Pr. No. G99TC13-0701, 2007.

(2) Schmitz, H. and Lyon, G., “Picking a better linear bearing”, Machine Design, Vol. 66 (1994), p.63-65.

(3) NSK Ltd. Precision Machine Parts, 1989.

(4) OHTA, H., “Vibration of linear guideway type recirculating linear ball bearings”,Journal of Sound and Vibration, Vol.235 (2000), p. 847-861.

(5) OHTA, H., “Sound of linear guideway type recirculating linear ball bearings”, ASME Journal of Tribology, Vol.121 (1999), p. 678-685.

(6) KASAI, S., TSUKADA, T. and KATO, S., “Precision linear guides for machine tools”

(in Japanese), NSK Technical Journal, 1987.

(7) KASAI, S., TSUKADA, T. and KATO, S., “Recent technical trends of linear guides”

(in Japanese), NSK Technical Journal, 1988.

(8) YE, J., IIJIMA, N., TASHIRO, F., HAGIWARA, S. and YAMADA, S., “Vibration of linear motion bearing”(in Japanese), Proceedings of Spring JSPE Meeting, (1988), p.

199-200.

(9) Schneider,M., Statisches und dynamisches Verhalten beim Einsatz, Liearer Schienfuhrungen auf Walzlagerbasis im Werkzeugmaschinenbau, Carl Hanser Verlag, Munchen, Wien., 1991

(10) Ju, S.H., Horng, T.L. and Cha, K.C., “Comparisons of contact pressures of crowned rollers”, Proc, mechE, Part J: J. Engineering Tribology, Vol.214 (2000), p.147-156.

(11) Nayak, L. and Johnson, K.L., “Pressure between elastic bodies having a slender area of contact and arbitrarily profiles”, Int. J of Mechanical Science, Vol.21 (1979), p.237-247.

(12) Lundberg, G., Elastiche Beruhreng zweier Halbraume, (Elastic Contact of Two Half Spaces), Forschung AUF dem Gebiete der Ingenieuwesens, 1939.

(13) Horng, T.L., Ju, S.H. and Cha, K.C., “A deformation formula for circular crowned rollers compressed between two plates”, Journal of Tribology-Transactions of the ASME, Vol.122 (2000), p.405-411.

(14) Horng, T.L., and Ju, S.H., “Stiffness of Arbitrarily Crowned Roller Compressed between Two Plates”, Proc, mechE, Part J: J. Engineering Tribology, Vol. 217 (2003), p.

375-384.

(15) Horng, T.L., “Analyses of Stiffness in an Arbitrarily Crowned Roller Compressed between Raceways”, Journal of the Chinese Society of Mechanical Engineerin, Vol.24, (2003), p.267-275.