Rubber Roller Design

8/20/2019 Rubber Roller Design

1/20

Copyright c 2008 ICCES ICCES, vol.7, no.1, pp.23-42

Analysis of Flexible Media Transport Characteristics of

Crown Roller and Its Application to DesignIizuka Yoichiro1, Hiroshi Yamaura

SummaryCrown rollers are often used in the flexible media transport mechanism consists

of a rubber layered roller and a steel roller. In this paper, the flexible media transportcharacteristics of crown roller is investigated theoretically. The purpose of thisstudy is to analyze the flexible media transport characteristics such as longitudinaldistributionof nip pressure, traction and transport velocity and so on, and to developthe design method of crown roller that has desired characteristics.

In the first step of this study, a numerical analysis method of the slip and trac-tion characteristics between a rubber-layered roller and a mated steel roller basedon the Boundary-Element Method was established. Under the assumption of aplane-strain, Green’s functions of a rubber roller with a rigid core based on polarcoordinates were analytically derived in our method. A contact problem of tworollers under normal and tangential loading forces was numerically analyzed us-ing derived Green’s function. Coulomb’s friction law, with a constant coefficientof friction, and no difference between the adhesion and the sliding coefficient of friction, was considered in the contact problem. The normal and tangential contactpressures, strain of rubber surface, nip width and indentation depth were iterativelycalculated so as to satisfy the given normal and tangential loading forces. Thetangential velocity ratio between the two rollers was also calculated for variousparameter values. It was confirmed that the numerical results agree well with the

experimental ones.In the second step of this study, an analytical approach that can estimate flexible

media transfer characteristics of crown rollers was developed. In the analysis, twotypes of crown rollers, i.e. a rubber roller with crown shape in its internal diameterand a steel roller with crown shape in its diameter, were examined. And basedon analysis results, a design approach of crown shape which can realize uniformlongitudinal distribution of nip pressure was proposed. Designed crown rollerswere fabricated and the performances of them were confirmed with experiment.

IntroductionFor the flexible media transport mechanism composed of long rollers, it is very

important to uniform longitudinal distribution of the nip pressure, traction, trans-port velocity, and so on. The deviation of longitudinal distribution of nip pressureis mainly subjected to the deflection of the roller shaft. Lots of methods that can

1Department of Mechanical and Control Engineering, Tokyo Institute of Technology. e-mail:[email protected]


2/20

24 Copyright c 2008 ICCES ICCES, vol.7, no.1, pp.23-42

reduce the deviation have been developed and used practically. One of the meth-

ods involves a technique to change the diameter of the roller along the longitudinalaxis, namely crown roller or crowning roller. Dimensions of the crown roller aredesigned experientially, because theoretical design method has not been revealedyet. Purpose of this study is to develop the analytical method of flexible mediatransport characteristics of crown roller and establish the theoretical design methodof the crown roller.

Some researchers have been focused on a contact problem in a flexible me-dia feeding mechanism. Soong and Li[8] considered a sheet in the nip of twocylinders and calculated pressure distribution for a normal load. In order to obtaintraction characteristics of a feeding mechanism analytically, normal and tangen-tial load should be considered. Soong and Li[9] proposed an analytical techniqueto calculate the traction characteristic between a rubber-layered roller and paper.They assumed a general stress function by the Fourier series in polar coordinates.In a nominal contact problem, their collocation method calculates the normal andtangential pressures at a contact area, tangential velocity ratios of two rollers andpaper. Kalker[10] proposed a numerical algorithm to solve a rolling contact prob-lem of viscoelastic multilayered cylinders with dry friction based on Bentall andJohnson[5]. Recently, this problem is focused upon and discussed in a researchworking group of the Japan Society for Precision Engineering. Wu et al. [11] inves-tigated the traction characteristics of a paper and rollers experimentally. Okamotoet al.[12] analyzed the same problem by using a commercial FEM software andinvestigated the effects of friction, non-linearity of elastic modulus and large de-formation, stress relaxation, rotating speed and the edge effect of a short roller as

well as a normal and tangential load, and showed the validity and usefulness of thenumerical analysis by experiment for short rollers. However, considering the factthat the actual feeding rollers in printers and copiers are longer than 200mm, it isalmost impossible to numerically analyze the effect of a non-uniform nip load andskewed angle of such long feeding rollers by FEM.

Accordingly, in the first step of this paper a numerical analysis of the slip andtraction characteristics between a rubber-layered roller and a mated rigid rollerbased on the Boundary-Element Method is presented. In order to investigate thebasic contact characteristics of a rubber roller, the situation of no flexible madiabetween the two rollers is considered. The Green’s function of a rubber-layeredroller based on polar coordinates is obtained under the plane-strain condition, and

then by using this Green’s function, the relation of load force and penetration depthof the two rollers is calculated. The validity of the calculated results are shown bythe experimental results.

In the second step, the crown roller is investigated based on the results of the


3/20

Flexible Media Transport Characteristics of Crown Roller 25

first step. Long rollers whose shaft deflection cannot be ignored have the longitu-

dinal distribution of the nip pressure. In this case, the rubber deformation is three-dimentional but it can be modeled with two-dimensional contact mentioned beforebecause the out-of-plane strain is much smaller than the plane strain. Thus, an an-alytical method that is based on the results of the two-dimensional contact analysisand can estimate the longitudinal distribution of nip pressure of the long rollersis developed. The analytical method is applied to a design method of the crownshape that can realize uniform longitudinal distribution of nip pressure. Two typesof crown rollers i.e. a rubber roller with crown shape in its internal diameter and asteel roller with crown shape in its diameterare, defined and design procedures of these rollers are shown. Dsigned crown rollers are fabricated and the performancesof them are confirmed with experiment.

Green’s Function DerivationBasic Equation and General Solution

Figure 1 depicts an analytical model of a rubber-layered steel roller and a matedsteel roller. The rubber is deformed by contact with the mated steel roller. BecauseYoung’s modulus of steel is larger than 105 times of that of rubber, the deformationof steel parts is ignored. Uniform contact along the axial direction of the rollers isassumed in this analysis and only the deformation of rubber in a section is consid-ered under the assumption of the plane-strain.

Figure 1: Analitical model of a rubberroller

Figure 2: Infinitely small element andstress


4/20


Figure 2 shows an infinitely small element and definition of stress. The equi-

librium equations of stress are given by

∂σ r ∂ r

+ 1

r

∂τ r θ ∂θ

+ σ r −σ θ

r = 0 (1)

2τ r θ

r +

∂ τ r θ ∂ r

+ 1

r

∂σ θ ∂θ

= 0 (2)

By substituting Hooke’s Law into stress σ r and τ r θ , these equations are repre-sented by normal and tangential deformation u,v.

κ

∂ 2u

∂ r 2 +

1

r

∂ u

∂ r −

u

r 2

+ β

r

∂ 2v

∂ r ∂θ −

κ 2−β 2

2(1 +ν ) +κ

1

r 2∂ v

∂θ

+ κ 2 −β 2

2(1 +ν )

1

r

∂ 2v

∂θ∂ r +

1

r 2∂ 2u

∂θ 2

= 0 (3)

β

r

∂ 2u

∂θ∂ r +

κ

r 2

∂ u

∂θ +

∂ 2v

∂θ 2

+ κ 2−β 2

2(1 +ν )

1

r 2∂ u

∂θ +

1

r

∂ v

∂ r −

v

r 2 +

1

r

∂ 2u

∂ r ∂θ +

∂ 2v

∂ r 2

= 0, (4)

where E and ν are a Young’s modulus and Poisson’s ratio of rubber, respectively.Notation κ and β are constants related to ν as follows: κ = 1−ν 2, β = ν (1 +ν ).

In order to solve these partial differential equations, Eqs.(3) and (4), u and v astwo Fourier series are defined as followings,

u ≡ G0(r ) +∞

∑k =1

Gck (r ) cos k θ +∞

∑k =1

Gsk (r ) sink θ (5)

v ≡ K 0(r ) +∞

∑k =1

K ck (r ) cos k θ +∞

∑k =1

K sk (r ) sin k θ (6)

Notation k represents number of harmonics. After substituting Eqs.(5) and (6)into Eqs.(3) and (4), an independent variable r is transformed into a new vari-able t with r = et in order to obtain six ordinary differential equations related with


5/20


G0(t ),K 0(t ),Gck (t ),Gsk (t ),K ck (t ), K sk (t ).d 2

dt 2−1

G0 = 0,

d 2

dt 2−1

K 0 = 0 (7)

κ d 2

dt 2−(κ +α k 2)

Gck +

(β +α )

d

dt − (κ +α )

kK sk = 0

κ d 2

dt 2−(κ +α k 2)

Gsk −

(β +α )

d

dt − (κ +α )

kK ck = 0

α d 2

dt 2−(κ k 2 +α )

K ck +

(β +α )

d

dt + (κ +α )

kGsk = 0

α d 2

dt 2−(κ k 2 +α )K sk −(β +α )

d

dt + (κ +α )kGck = 0

(8)

The notation α is defined as

α = κ 2−β 2

2(1 +ν ).

By solving Eqs.(7), general solutions of G0(t ) and K 0(t ) are obtained as follows:

G0(t ) = C G01et +C G02e

−t

K 0(t ) = C K 01et +C K 02e

−t

, (9)

where C G01,C G02,C K 01 and C K 02 are constants calculated from boundary condi-tions. Eq.(8) is rather complex to be solved. Primarily, four eigen values of Eqs.(8)

are calculated asλ k 1 ∼ λ k 4 = k + 1, −k −1, k −1, −k + 1. (10)

Considering λ 13 = λ 14 = 0 for k = 1, the general solutions of Gc1(t ),Gs1(t ),K c1(t ),K s1(t ) of the forms are obtained.

Gc1(t ) =2

∑ j=1

A1 j exp(λ 1 jt ) + A13 + A14t

Gs1(t ) =2

∑ j=1

B1 j exp(λ 1 jt ) + B13 + B14t

K c1(t ) =2∑

j=1

η1 j B1 j exp(λ 1 jt ) + ( B13 +η13 B14) + B14t

K s1(t ) = −2

∑ j=1

η1 j A1 j exp(λ 1 jt )−( A13 +η13 A14)− A14t

(11)


6/20


where

η11 = − 5−4ν −1 + 4ν

, η12 = 1, η13 = 13−4ν (12)

And general solutions of Gck (t ),Gsk (t ),K ck (t ) and K sk (t ) for k = 2,3, . . . are derivedas follows:

Gck (t ) =4

∑ j=1

Ak j exp(λ k jt )

Gsk (t ) =4

∑ j=1

Bk j exp(λ k jt )

K ck (t ) =

4

∑ j=1ηk j Bk j exp(λ k jt )

K sk (t ) = −4

∑ j=1

ηk j Ak j exp(λ k jt )

(13)

where

ηk 1 = − k + 4(1−ν )

k −2(1−2ν ), ηk 2 = 1

ηk 3 = −1, ηk 4 = k −4(1−ν )

k + 2(1−2ν )

(14)

And Ak j and Bk j (k = 1,2, . . .; j = 1 ∼ 4) are also constants calculated from theboundary conditions.

Boundary Conditions at the Inner RadiusThe boundary conditions at the inner radius of the rubber roller, Rin,are u( Rin,θ ) =

v( Rin,θ ) = 0. Subsequently, the following equations are obtained

G0(ln Rin) = K 0(ln Rin) = 0 (15)

Gck (ln Rin) = Gsk (ln Rin) = 0

K ck (ln Rin) = K sk (ln Rin) = 0 (k = 1,2, · · ·) (16)

Boundary Conditions at the Outer RadiusThe boundary conditions at the outer radius of the rubber roller, Rout , are ex-

pressed by two sets of Fourier series for two loading conditions.


7/20


First, for the normal unit force, the boundary conditions are :

σ r ( Rout ,θ ) = limd θ →0

1 Rout d θ

= 1

Rout δ (θ )

= 1

2π Rout +

1

π Rout

∞

∑k =1

cos k θ (17)

τ r θ ( Rout ,θ ) = 0 (18)

where δ (θ ) is the Dirac’s δ function.

Second, for the tangential unit force, the boundary conditions are :

σ r ( Rout ,θ ) = 0 (19)τ r θ ( Rout ,θ ) =

1

2π Rout +

1

π Rout

∞

∑k =1

cos k θ . (20)

Solutions of the Green’s FunctionThe boundary conditions at the outer radius, shown in Eqs.(17), (20), are ex-

pressed not by deformations but by stresses. However, these stresses are theoreti-cally related to deformations.

σ r = E

κ 2−β 2

κ ∂ u

∂ r +β

u

r +

1

r

∂ v

∂θ

(21)

τ r θ =

E

2 (1 +ν )1

r

∂ u

∂θ +

∂ v

∂ r −

v

r

(22)

By substituting Eqs.(5) and (6) to Eqs.(21) and (22) and considering the harmonicbalance with the boundary conditions of Eqs.(17) and (18) or Eqs.(19) and (20), weobtain the equations of the Fourier coefficients in Eqs.(5) and (6). Ensuingly, it isfinally determined that the Fourier coefficients by using the boundary conditions atthe inner radius in Eqs.(15) and (16).

Contact ProblemIn this section, the way to solve contact problem is shown.

Analytical Model and AssumptionFigure 3 depicts an analytical model of a flexible media transport mechanism

consisted of a rubber-layered steel roller and a steel roller. The rubber is deformedby contact with the paper supported by the steel roller. Because Young’s modulusof steel is larger than 105 times of that of rubber, the deformation of steel parts isignored. O− XZ is a Cartesian coordinate system whose origin is set on an un-deformed surface of the rubber. The coordinates, o− r θ , denote polar coordinates


8/20


whose origin is the center of the rubber roller. The rotation direction of each roller

has been shown in the figure. Negative and positive X corresponds to the inlet andoutlet of rotation, respectively.

Figure 3: Contact model

In the previous section, Green’s functions of the cylindrical rubber were de-rived analytically under the plane strain assumption. In this section, the followingassumptions are further considered.

• Bending stiffness of the flexible media and deformations of the flexible media

in the nip can be ignored.• Tension T works on the flexible media directory and no break torque workson the steel roller.

In this section, contact normal and tangential pressure distributions, stick andslip area, strain of the rubber surface against the normal force F and the tension T are calculated. The rigid core of the rubber roller is fixed in space but rotatory. Thesteel roller can move along the Z-axis and the penetration depth is δ Z (> 0). Thetension of the paper −T is acting on the rubber surface so that a point on the freerubber surface at X = 0 moves to X = δ X


9/20


denote the normal and tangential deformation due to a tangential pressure q(θ ),

respectively. These deformations are represented in the following equations usingGreen’s functions.

U p(ψ ) =

a−b

u p( Rout ,ψ −θ ) p(θ ) Rout d θ (23)

V p(ψ ) =

a−b

v p( Rout ,ψ −θ ) p(θ ) Rout d θ (24)

U q(ψ ) =

a−b

uq( Rout ,ψ −θ ) q(θ ) Rout d θ (25)

V q(ψ ) = a

−b

vq( Rout ,ψ −θ ) q(θ ) Rout d θ (26)

We assume the linear elastic deformation in this analysis because the nonlineardeformation should not be utilized in the actual mechanism from the viewpoint of durability of the rubber roller.

Formulation of Contact AnalysisThe condition equations of deformation, normal pressure, tangential pressure

and boundary of the contact area, are shown in this subsection.

Rout

R in

ψ

ψ

Rss

+( s+Tp)(1-cosψ s)+Rs(1-cosψ s)

Rout(1-cosψ )

Figure 4: Initial gap

A steel roller whose outer radius is Rs is considered. The initial gap of thedirection of Z is defined as shown in Fig.4. A deformation conditionof the direction

of Z was derived.

−{U p(ψ ) +U q(ψ )}cosψ +{V p(ψ ) +V q(ψ )}sinψ

= δ Z − Rout (1−cosψ )− Rs(1−cosψ s) +ε θ Rout ψ sinψ , (27)


10/20


where U p,V p,U q,V q are deformations on the rubber surface given by Eqs.(23)∼(26).

And ψ s is defined as the following equation.

ψ s = sin−1

Rout

Rs sinψ

(28)

A deformation condition of the direction of X should be considered in the stickarea. Coulomb’s frictional low, with a constant coefficient of friction, µ , and no dif-ference between the adhesion and the sliding coefficient of friction between the pa-per and the rubber surface is assumed. The absolute value of the tangential pressureq doesn’t exceed µ p in the stick area and a deformation condition of the directionof X is given,

{U p(ψ ) +U q(ψ )} sinψ +{V p(ψ ) +V q(ψ )}cosψ

= δ X + ε θ Rout ψ cosψ . (29)

On the other hand, in the slip area, an alternative condition equation should beconsidered.

|q(θ )|= µ | p(θ )| (30)

From the equilibrium conditions of normal and tangential pressure, the normaland tangential loading forces are given by

F =

a−b{− p(θ ) cosθ + q(θ ) sinθ } Rout d θ , (31)

T =

a−b{ p(θ ) sinθ + q(θ ) cosθ } Rout d θ . (32)

Boundary conditions of the contact area are given as follows.

p(−b) = p(a) = 0 , q(−b) = q(a) = 0 (33)

In the calculation procedure, normal and tangential pressures, p and q, aredefined on 50 points in the contact area and p, q, δ Z , δ X , ε θ , a and b are iterativelycalculated by using Eqs.(27), (33).

Calculated Result

Calculated results of the relation of load force and penetration depth are mainlydiscussed below.

The examples of the relation of load force and penetration depth obtained bytheoretical analysis are shown in Fig.5. In the calculation, the outer radius of eachroller is 15mm, Young’s Modulus of rubber are 5.57MPa and 1.35MPa, Poisson’s


11/20


ratio of rubber ν is 0.49, friction coefficients µ is 0.5 and the normal loading force

F is changed from 100 to 700N/m. In all calculation, tangential loading force T is 0. In this calculation, Young’s Modulus of rubber E r , rubber thickness T r , loadforce F were chosen as variable parameters.

0

100

200

300

400

500

600

700

800

0 0.0002 0.0004 0.0006

F [ N / m ]

z [m]

3mm 5.5mm

8mm

Tr=

(a) 30Hs

0

100

200

300

400

500

600

700

800

0 0.0001 0.00015 0.0002

F [ N / m ]

z [m]

0.00005

3mm

5.5mm

8mm Tr=

(b) 70HsFigure 5: Nonlinear spring characteristics of rubber roller

Weight

Driven

Velocity

control

Rubber

Roller

Steel

Roller

Figure 6: Experimental setup of short roller

It is found from Fig.5 that the thinner the rubber is the harder the rubber rollerbehaves. And, each plot can well approximate with quadratic function that throughsorigin. For example, the case of T r = 5.5mm, E r = 5.57MPa, approximate quadratic

function isF = 3.424×106δ z + 1.253×10

10δ 2 z .


12/20


Preliminary Experiment and Discussion

Experimental Apparatus and ConditionsIn order to confirm the validity of our calculated results, an experimental setupshown in Fig.6 was prepared. The radius and length of each roller is 15mm and40mm, respectively. The bearing span of rollers is 55mm. The steel roller is adrive roller that is actuated by a motor with a velocity controller. The rubber rolleris a driven roller that rotates freely and is supported in a vertical plane with guideshafts. The normal loading force is given to the rollers by weights and decidesthe penetration depth. In order to obtain the relation of load force and penetrationdepth, the normal loading force F was changed from 100 to 700N/m. The pen-etration depth was measured as the displacement of rubber roller unit by a CCDcamera and monitor. All experiments were performed with rotational speed of thesteel roller of 10rpm.

Six silicon rubber rollers are prepared for experiments. The thickness of rub-bers are 3.0mm,5.5mm and 8.0mm. The hardness of the rubbers are s =70 Hs ands =30 Hs and the value of Young’s Modulus was calculated from the hardness withGent’s equation[14].

E (MPa) = 9.8×10−2 s + 7.31

0.0456(100−s) (34)

Comparison Between Calculated and Experimental ResultsFigure 7 indicates the relation of load force and penetration depth obtained

experimentally. Eight experiments were achieved for one loading condition and the

maximum, average and minimum values are shown in the figures. It is noted thatthe calculation results agree well with experimental ones.

Analysis of the Longitudinal Distribution of the Nip PressureLong rollers whose shaft deflection cannot be ignored have the longitudinal

distribution of the nip pressure as shown in Fig.8. In this case, the rubber defor-mation is three-dimentional but it can be modeled with two-dimensional contactmentioned before because the out-of-plane strain is much smaller than the planestrain. In this section, an analytical method that is based on the results of the two-dimensional contact analysis and can estimate the longitudinal distribution of nippressure of the long rollers is developed.

Two-dimensional model of rubber deformation leads a simple model with par-

allel independent nonlinear springs shown in Fig.9. The nonlinear characteristicsof springs were obtained in the two-demensional analysis shown in Section . Thestatic deformation of the rubber roller shaft zr ( y) should be considered in the anal-ysis of the longitudinal distribution of the nip pressure. When the longitudinaldistributionof the nip pressure F ( y) is given, zr ( y) is calculated from the following


13/20


0

100

200

300

400

500

600

700

800

0 0.0002 0.0004 0.0006

F [ N / m ]

z [m]

NumericalAverage

MinMax

(a) 30Hs, T r = 3 mm

0

100

200

300

400

500

600

700

800

0 0.00005 0.0001 0.00015 0.0002

F [ N / m ]

z [m]

NumericalAverage

MinMax

(b) 70Hs, T r = 3 mm

0

100

200

300

400

500

600

700

800

0 0.0002 0.0004 0.0006

F [ N / m ]

z [m]

NumericalAverage

MinMax

(c) 30Hs, T r = 5.5 mm

0

100

200

300

400

500

600

700

800

0 0.00005 0.0001 0.00015 0.0002

F [ N / m ]

z [m]

NumericalAverage

MinMax

(d) 70Hs, T r = 5.5 mm

0

100

200

300400

500

600

700

800

0 0.0002 0.0004 0.0006

F [ N

/ m ]

z [m]

NumericalAverage

MinMax

(e) 30Hs, T r = 8 mm

0

100

200

300400

500

600

700

800

0 0.00005 0.0001 0.00015 0.0002

F [ N

/ m ]

z [m]

NumericalAverage

MinMax

(f) 70Hs, T r = 8 mmFigure 7: Experimental results of relation of load force and penetration depth

equation.

E s I sd 4 zr ( y)

dy4 = F ( y), (35)

where E s and I s = (π R4c)/4 are an elastic modulas and the second moment of area

of the rubber roller shaft, respectivelly.

On the other hand, the static defromation of the steel roller was ignored inthe analysis, because it is thicker than the rubber roller shaft. In the calculation


14/20


F L [N] F R [N]

F L [N] F R [N]

φ 3 0

2 R a

2 R b

2Ra

2 R b

Lb Lc

T r

La La Lb

Steel

Rigid Body

Rubber

F [N/m]

φ 3 0

2 R c

z

o

y

δ s (y)

z (y)

Figure 8: Calculation model

Steel roller

Rubber roller shaft

Rubber

Figure 9: Nonlinear spring model

0

200

400

600

800

1000

0 0.06 0.12 0.18 0.24

F [ N / m ]

y [m]

30Hs 8mm30Hs 3mm70Hs 8mm70Hs 3mm

Figure 10: Longitudinal distribution of nip pressure


15/20


procedure, the stiffness of the rubber roller shaft can be calculated from Eq.(35) by

FEM and utilized with nonlinear stiffness of rubber.Examples of calculated longitudinal distribution of the nip pressure of straightrollers were shown in Fig.10. Load force acting on each side was assumed to be58.8N in these calculations. More than 10% deviation of the longitudinal distribu-tion of the nip pressure is observed even in the bast case of the examples.

Crown DesignIn this section, the way to uniform the longitudinal distribution of nip pressure

using crown roller is shown.

A crown roller generally means a steel roller with crown shape in its diameter.However, other types of crown rollers, i.e. a rubber roller with crown shape in itsinner or outer diameter were able to assumed. Because it is difficult to fabricate

rubber with high accuracy, a rubber roller with crown shape in its outer diameterwas ignored. Thus, two types of crown roller were discussed. The dimensions of each type of crown roller were calculated with a computer program whose flowchart is shown in Fig.11.

Start

End

Yes

No

Figure 11: Flow chart of crown calculation

Crown of Steel Roller

In this subsection, design procedure of crowned steel roller is shown.From the two-dimensional analysis, nonlinear spring characteristics of the rub-

ber roller is derived as

F ( y) = b1δ z( y) + b2δ z( y)2 (36)


16/20


Notations b1 and b2 were resulting coefficients derived from nonlinear spring char-

acteristics of the rubber roller. From this equation, the following equation is ob-tained.

δ z( y) =−b1 +

b21 +4b2F ( y)

2b2(37)

In the case of straight rollers, each position of the rubber surface should movea constant distance by contact. The rubber roller shaft deflection is a function of y, then the penetration depth of rubber roller should be a function of y. This isthe reason of the nonuniform longitudinal distribution of nip pressure. In order toobtain uniform longitudinal distributionof nip pressure, constant penetration depthof rubber roller required. In the design procedure, a reference penetration depth ischosen and the diameter of the steel roller at y, Rs( y), is changed iteratively so that

the penetration depth at y coincides with that of the reference point yr .

Rs( y)new = Rs( y)old −δ z( y) +δ z( yr ) (38)

0.0147

0.0148

0.0149

0.015

0.0151

0 0.06 0.12 0.18 0.24

R s [ m ]

[m]

3mm5.5mm

8mm

Figure 12: Example of crown shape of steel roller

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0 0.06 0.12 0.18 0.24

R

[ m ]

y [m]

30Hs 8mm30Hs 3mm70Hs 8mm70Hs 3mm

i n

Figure 13: Example of crown shape of rubber roller shaft

Figure 12 shows an example of crown shape derived from calculation. Thecrown shape actually agrees with the rubber roller shaft deflection loaded staticuniform load force. It is found from Fig.12, calculated crown height Rsmax − Rsminis less than 20µ m. Thus, it can be concluded that uniform longitudinal distributionof nip pressure is hardly achieved practically with the crowned steel roller.

Crown of Rubber RollerIn this subsection, design procedure of crowned rubber roller shaft is shown.

Thinner rubber has harder spring characteristics. Using this property, longitu-dinal distribution of nip pressure could be uniformed. Thus, the rubber should bethickened where load force is strong, or be thinned where load force is weak.


17/20


As mentioned before, the rubber roller has nonlinear spring characteristics

shown in Eq.(36). And the coefficients of b1 and b2 in this equation are functionsof the inner radius of the rubber roller. These coefficients were obtained for var-ious inner radius by two-dimensional analysis and derived interpolation functionsof them.

bk (r ) = B+1

∑ j=1

C k j R j−1in (k = 1,2), (39)

where Rin is a radius of the rubber roller shaft and B is the number of calculatedcase. Thus,

F ( y) = b1( y)δ z( y) + b2( y)δ z( y)2

= B+1

∑ j=1

δ z( y)C 1 j +δ z( y)C 2 j Rin( y) j−1 (40)From this equation, the radius of the rubber roller shaft at y that gives the samenip pressure with that of the reference point yr was callculated iterativelly by thefollowing equation.

Rin( y)new = F ( yr )−F ( y)

F ( y) + Rin( y)old (41)

where

F ( y) = B+1

∑ j=1

( j−1)δ z( y)

C 1 j +δ z( y)C 2 j

Rin( y)

j−2old

Examples of crown shape of the rubber roller shaft are shown in Fig.13. Inthese cases, the roller end was chosen as the reference position. Even in the caseof soft and thin rubber, the calculated crown height Rinmax − Rinmin is about 300µ m.It can be concluded that uniform longitudinal distribution of nip pressure will beachieved practically with the crowned rubber roller shaft.

Experiment and DiscussionIn this section, the experiment of crowned rubber roller shaft is shown.

Experimental Apparatus and ConditionsOnly one type crown rollers were fabricated that were rubber rollers with crown

shape in its internal diameter. One of crown rollers proparties are 70Hs hardness

rubber, the thickness of rubber is 3mm at the edge of roller. Those of the otherone are 30Hs hardness rubber, the thickness of rubber is 3mm at the edge of roller.These rollers were designed for 58.8N loading force for each side of rollers.

In order to confirm the validity of our calculated results, an experimental setupshown in Fig.14 was prepared. To measure the longitudinal distribution of nip


18/20


Figure 14: Test bench of Crown roller

0

0.25

0.5

0.75

1

0 0.25 0.5 0.75 1

F

/ F m a x

y / L

with crown experimentalstraight numerical

(a) 30Hs

0

0.25

0.5

0.75

1

0 0.25 0.5 0.75 1

F

/ F m a x

y / L

with crown experimentalstraight numerical

(b) 70HsFigure 15: Experimental result of longitudinal distributionof nip pressure of crownroller

pressure, pressure measuring film called PRESCALE was used. The loading forceis 58.8N for each side of rollers. Crown roller is drived by servo motor of 10rpm.

Straight steel roller rotates freely.Comparison Between Calculated and Experimental Results

Because of PRESCALE characteristics, measured pressure is very noisy. Sothe average of pressure of its driving direction is derived from measured data. AndPRESCALE can’t measure precise puressure, the averaged pressure was normal-


19/20


ized with its maximum value. The result of these calculation of experimental data is

called normalized pressure distribution. If longitudinal distribution of nip pressureis uniformed, normalized pressure distribution is flat and its value is 1.

Figure 15 shows examples of the experimental results. Compared to the cal-culation results of normalized pressure distribution of straignt rollers, our crownroller gives well uniformed pressure distribution.

Table 1: Roller demensions

a b c

L[mm] 7 93 240R[mm] 5 7 9.5

ConclusionIn this paper, firstly, two-dimensional contact problem was solved theoretically

with Green’s functions of a rubber roller with a rigid core. And the relation of loadforce and penetration depth was obtained experimentally and theoretical resultswere confirmed by experimental results.

Secondly, the way to obtain crown shape that can uniform longitudinal dostri-bution of nip pressure was shown. In this design procedure, the results of two-dimensional contact problem was utilized. Experimental results were also com-pared with theoretical ones and discussed.

The results of this study are summarized as follows:

1. The way to obtain Green’s function was shown.2. The way to solve two-dimentional contact problem was shown.

3. Calculated results were confirmed with experiments of the relation of loadforce and penetration depth.

4. The method to obtain the londitudinal pressure distribution of nip pressureof a long rollers was revailed.

5. The dedign procedure to obtain crown shape based on the calculated relationof load force and penetration depth was shown.

6. Designed crown rollers were fabricated and the performances of them wereconfirmed with experiment.

References

1. Fujimura, H., and Ono, K., 1996, “Analysis of paper motion driven byskew-roll paper feedinf system," Transactions of JSME , Vol.62(C), No.596,pp.1354-1360.(in Japanese)


20/20


2. Hertz, H., 1881, “Über die Berührung fester elastischer Koerper," J. Reine

Angew. Math., No.92, pp.156-171.3. Hannah, M., 1951, “Contact Stress and Deformation in a thin Elastic Layer,"

Quart. Journ. Mech. And Applied Math., Vol.4, Pt.1, pp.94-105.

4. Parish, G.J., 1958, “Measurements of pressure distribution between metaland rubber covered rollers," British Journal of Applied Physics, Vol.9, pp.158-161.

5. Bentall, R.H., and Johnson, K.L., 1967, “Slip in the Rolling Contact of TwoDissimilar Elastic Rollers," International Journal of Mechanical Sciences,Vol.9, pp.389-404.

6. Hahn, H.T., and Levinson, M., 1974, “Indentation of an Elastic Layer(s)Bonded to a rigid Cylinder - I. Quasistatic Case without Friction," Interna-tional Journal of Mechanical Sciences, Vol.16, pp.485-502.

7. Gupta, P.K., and Walowit, J.A., 1974, “Contact Stresses Between an ElasticCylinder and a Layered Elastic Solid," Transactions of the ASME , Series F,Vol.97, No.2, pp.250-257.

8. Soong, T.C., and Li, C., 1981, “The steady rolling contact of two elasticLayer bonded cylinders with a sheet in the nip," International Journal of Mechanical Science, Vol.23, pp.263-273.

9. Soong, T.C., and Li, C., 1981, “The rolling contact of two elastic-Layer-covered Cylinders driving a loaded sheet in the nip," Transaction of the ASME, Jounal of Applied Mechanics, Vol.48, pp.889-894.

10. Kalker, J.J., 1991, “Viscoelastic Multilayered Cylinders Rolling with DryFriction," Transaction of the ASME, Journal of Applied Mechanics, Vol.58,No.3, pp.666-678.

11. Wu, J., Nakazawa, M., Kawamura, T.,Baba, H., Kamimura, H., Tanaka, N.,and Miyahara, K., 1999, “Study on Elastic Deformation Properties of theBlanket in Offset Printing," JSME International Journal, Series A, Vol.42,No.3, pp.388-393.

12. Okamoto, N., Ohtani, K., Misawa, K., Yoshida, K., 2001, “Study on Ve-locity Characteristics and Mechanics of Paper Feeding with Rubber-CoveredRoller Drive," Transaction of the JSME , Vol.67(C), No.654, pp.475-482. (in

Japanese)13. Johnson, K.L., 1985, Contact Mechanics, Cambridge University press

14. A.N.Gent, 1958, “On the relation between indentation hardness and Young’smodulus," Transactions of the Institution of the Rubber Industry,34

Documents

Rubber Roller Design