New Tribological Ways Part 8 doc

The normal force to a small element of the inner belt at angle θ is denoted as dN b, which can be written as 2 2 b b Making use of T4’ and T4”, the normal forces of inner belt for an eac

A Comparison of the Direct Compression Characteristics of Andrographis paniculata,

Eurycoma longifolia Jack, and Orthosiphon stamineus Extracts for Tablet Development 229

Fig 5 (b) Walker plots of tablets prepared from Eurycoma longifolia Jack, Andrographis

paniculata and Orthosiphon stamineus for 1.0 g of feed powders

lastly the Andrographis paniculata extract powder The high density gave the high value of

the tensile strength, which was related to the reduction in the void space between particles

in the powders during tablet formation with respect to the values of the slopes, which decreased as the tensile strength increased

compress, and it underwent significant particle rearrangement at low compression pressures, resulting in low values of yield pressure The compression characteristics of the

Eurycoma longifolia Jack powder were consistent when validated with all of the models used

Another significant finding showed that the characteristics of 0.5 g of feed powder are better than for 1.0 g of feed powder, as proven from the tensile strength test; hence a more coherent tablet can be obtained Thus, herbal parameters are superior when screening extract powders with the desired properties, such as plastic deformation This study also validated the use of Heckel, Kawakita and Lüdde, and Walker model parameters as acceptable predictors for evaluating extract powder compression characteristics

5 Acknowledgements

This work was supported by a research grant from the Ministry of Higher Education

Malaysia, Fundamental Research Grant Scheme with project number: 5523035 and

Universiti Putra Malaysia (UPM) Research University Grant Scheme with project number:

91838 Some of the authors were sponsored by a Graduate Research Fellowship from the

UPM

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Part 3

Tribology and Low Friction

The self-locking of belt may occur even in the case where a belt is wrapped on an axis two or more times The second purpose of this chapter is to present the frictional property of belt wrapped on an axis two and three times through deriving the formulas corresponding to an each condition Making use of this self-locking property of belt, a belt-type one-way clutch can be made (Imado, 2010) The principle and fundamental property of this new clutch are described

As the last part of this chapter, the frictional property of flexible element wrapped on a hard body with any contour is discussed The frictional force can be calculated by the curvilinear integral of the curvature with respect to line element along the contact curve

2 Theory of belt buckle

Notation

C Magnification factor of belt tension

F Frictional force, N

Fij = Fji Frictional force between point P i and P j , N

L Distance between two cylinder centers, m

N Normal force of belt to surface, N

Nij= Nji Normal force of belt between point P i and P j , N

Pi Boundary of contact angle

R Radius of main cylinder, m

Ti Tension of belt in i’th interval, N

r Radius of accompanied cylinder, m

μ Coefficient of friction for belt-cylinder contact

μ b Coefficient of friction for belt-belt contact

θi Angle of point P i

θij =θ ji Contact angle between P i and P j

2.1 Friction of belt in belt buckle

Figure 1 (a) shows a cross sectional view of a belt buckle and a belt wrapped around the two

cylindrical surfaces T1 and T4 (T1>T 4 ) are tensions of the belt at both ends There is a

double-layered part where the belt is wrapped over the belt Figure 1 (b) shows the enlarged

view around the main axis For simplicity, the thickness of the belt was neglected

According to the theory of belt friction, following equations are known for belt tensions of

T1 , T2 and T3 (Joseph F Shelley, 1990)

T4’ and T4” are of inner belt tension at P1 and P2 respectively The normal force to a small

element of the inner belt at angle θ is denoted as dN b, which can be written as

2

2

b b

Making use of T4’ and T4”, the normal forces of inner belt for an each section are expressed as

(a) Belt buckle (b) Enlarged view

Fig 1 Mechanical model of belt buckle and enlarged veiw around main axis

Frictional Property of Flexible Element 237

2 1 6

The frictional force F12 acting on the inner belt is composed of two forces denoted as F 12in

and F 12out The frictional force F 12in is acting on the cylindrical surface, which is generated by

the normal forces dN b anddN12 The normal force dN b is exerted from the outer belt The

other normal force dN12 is generated by the inner belt tension So, F 12in is given by

Substituting Eq (1) into Eq (11) to eliminate T2 gives

56 12

2.2 Property of formulas of belt buckle

The validity of Eqs (13) and (14) might be checked by supposing an extreme case of either

μ=0 or μ b =0 Substituting μ=0 into Eq (13) gives

12 12

2

b b

Substituting μ b =0 into Eq (15) or substituting μ=0 into Eq (16) gives T1=T4 Substituting

θ = into Eq (13) to remove the double-layered segment on the ratio of belt tension yields

the conventional equation of belt friction

34 56

Equation (17) is also obtained by substituting θ12= into Eq (16) This means that the ratio 0

of belt tension is magnified by the factor C

34 25

12

11

due to the double-layered segment even in the case of μ b=0 As far as these inspections are

concerned, there is no contradiction in Eq (13) As Eqs (13), (15) and (16) are of fractions,

the factor of T4 might become infinity meaning T4/T1=0 This fact virtually implies the

occurrence of self-locking Figure 2 shows the relation of μ b and θ12 satisfying eμ θb 12 = in 2

Eq (15) Self-locking occurs in the region above this curve where eμ θb 12 > On the other 2

hand, in the region below this curve, self-locking does not occur In the case of μ=0, the

equilibrium of moment of belt tension about O in Fig 1 gives

Frictional Property of Flexible Element 239

60 90 120 150 180

In the locking state with μ=0, T4=0 so that T1=2T2=2T3 It means that belt tension T1 is halved

to T2 by the belt-belt friction

As the angle of double-layered segment θ12 is determined by the geometry of the buckle,

some calculations were carried out to know the properties of Eq (13) and Eq (15) providing

r/L=R/L=1/4 The direction of belt tension T1 and T4 were assumed to be the same direction

for simplicity Results are shown in Figs 3 and 4 Figure 3 corresponds to the Eq (15) where

110100

Fig 3 Change of belt tension ratio with unfolding angle ζ in the case of μ=0 Belt tension

ratio increases greatly with an increment of the coefficient of friction μ b especially in the

vicinity of locking condition It is very sensitive to angle ζ

20 40 60 80 100 120 140 1601

Unfolding angle of buckle ζ, deg

Fig 4 Change of belt tension ratio with unfolding angle ζ in the case of μ=μ b Belt tension

ratio increases greatly with an increment of the coefficient of friction

Fig 5 Change of belt tension ratio with unfolding angle ζ in the case of μ=0 The ratio of belt

tension changes according to Eqs (13) or (15)

the coefficient of friction is μ=0 The ratio of belt tension increases with an increment of the

coefficient of friction μ b It increases greatly when it approaches the locking condition

Figure 4 shows some results obtained by Eq (13) providing μ=μ b The ratio of belt tension

becomes far bigger than the that of Fig 3

Some experiments were carried out to verify the validity of Eq (15) by wrapping a belt

around the outer rings of rolling bearings to realize the condition of μ=0 Belt tension T1 was

applied by the weight Belt tension T4 was measured by the force gauge Figure 5 shows the

results Experimental data are almost on the theoretical curves As predicted by the Eq (15),

self-locking was confirmed for the belt with μ b =0.5 in the region of ζ<10˚ where eμ θb12 > 2

Frictional Property of Flexible Element 241

2.3 Calculation of arm torque

Figure 6 (a) shows the mechanical model of belt buckle (Imado, 2008 a) Figure 6 (b) shows a

three-dimensional model of the buckle The arm of the buckle rotates around the point O2

The angle of arm is denoted by α The intersection angle of the line O-O2 and O2-O1 is

denoted by β From geometrical consideration, the angle β is given by

ζ , the angle of center line O-O1, can be calculated from the arm angle α by Eq (24) Note

the angle ζ is equal to α when L1 becomes 0

The moment of the arm about point O2 due to belt tensions T2 and T3 is expressed by

where c2 and c3 are geometrical variables that can be calculated from the position of contact

boundaries P2, P3, P4 and P5 Dividing the arm torque M with RT1, torque due to belt tension

T1 about point O, gives non-dimensional moment N

3 2

T T

Figure 7 shows some examples of non-dimensional torque N For simplicity, the coefficients

of friction were taken to be μ=μ b The non-dimensional torque N decreases to be negative

value with decrement of arm angle α It means an occurrence of directional change in arm

torque This negative torque acts so as to hold the arm angle in a locking state without any

locking mechanism The angle where arm torque N becomes 0 is denoted by αC It depends

on the geometry of buckle and the coefficients of friction μ and μ b Making use of Eqs (13)

and (24), the fraction of belt tension can be calculated Figure 8 shows some results The

fraction of belt tension, T4/T1, decreases with arm angle α It becomes 0 at α α= L

According to Eq (13), the fraction of belt tension T4/T1 becomes negative when arm angle α

becomes less than αL, α α< L The physical meaning of negative value in the fraction of belt

tension is that the belt tension T4 should be compressive so as to satisfy the equilibrium

condition of the force But a belt cannot bear compressive force so that negative value in the

fraction of belt tension is actually unrealistic It means the belt was locked with the buckle

The angle αL becomes larger with an increment of the coefficients of friction As the

coefficient of friction is generally greater than 0.15, the locking condition is easily satisfied

Once the locking condition is satisfied, the belt is dragged into the buckle with a decrement

of arm angle α Then the belt tension becomes greater

Fig 6 Mechanical model of belt buckle to calculate arm torque and 3D model

3 Theory of belt friction in over-wrapped condition

3.1 Friction of belt wrapped two times around an axis

Figure 9 shows a mechanical model (Imado, 2008 b) The point P i (i=1, 2, 3) is a boundary of

contact and T i (i=1, 2, 3, 4) is tension of the belt Symbol θ i denotes the angle of point P i The

belt is over-wrapped around the belt in the range from P 1 to P 2 denoted by θ 1 The axis x is

taken so as to pass through the point P 2 , which is an end of the belt T 1 is bigger than T 4 T 4 is

an imaginary belt tension There is no contact from P 2 to P 3 due to the thickness of the

belt-end According to the theory of belt friction (Joseph F Shelley, 1990), analysis starts with the

conventional equation

1 1

Frictional Property of Flexible Element 243

Fig 7 Non-dimensional arm toruque N decreases with arm angle α

Fig 8 Fraction of belt tension T4/T1 decreases with arm angle α

Fig 9 Mechanical model of belt wrapped two times around an axis

The belt tension T 2 or T3 can be expressed by the belt tension T 3 ’, where T 3 ’ is inner belt

tension at the point P 1 as shown in Fig 9

The inner belt is normally pressed onto the cylinder by the outer belt The normal force to a

small segment of the inner belt at angle θ denoted by dN b is

2

b b

On the other hand, the normal force is also generated by inner belt tension itself The normal

force exerted on the cylinder between Pi and Pj is denoted by Nij Normal force acting to a

small segment of the cylinder at angle θ is given by

Then, making use of Eqs (31) and (32), the frictional force between the inner belt and

cylinder denoted by F 12in is given by

Denoting the radius of cylinder by r and neglecting the thickness of the belt, the equilibrium

equation of moment of the cylinder is