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. Constants Material Feed powder (g) W B R 2 value Andrographis paniculata -0.16±0.001 2.00±0.006 0.971 Eurycoma longifolia Jack -0.24±0.001 2.33±0.056 0.984 Orthosiphon stamineus 0.5 -0.14±0.056 2.24±0.243 0.957 Andrographis paniculata -0.14±0.002 1.99±0.000 0.948 Eurycoma longifolia Jack -0.31±0.001 2.66±0.053 0.992 Orthosiphon stamineus 1.0 -0.27±0.105 2.67±0.421 0.928 Table 4. The Walker model 4. Conclusion This study on direct compression characteristics of selected Malaysian herb extract powders helped to deduce and understand some of the important principles of tablet development. The Eurycoma longifolia Jack extract powder was the easiest of the three herb powders to 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. New Tribological Ways 230 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. 6. References Abdul Aziz, R., Kumaresan, S., Mat Taher, Z. & Chwan Yee, F.D. (2004). 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Physical Properties and Compact Analysis of commonly Used Direct Compression. Pharmaceutical Science Technology, 4(4), 62, ISSN 1530-9932. (available from http://www .apppspharmscitech. org - Accessed 21/6/2008). Part 3 Tribology and Low Friction 12 Frictional Property of Flexible Element Keiji Imado Oita University Japan 1. Introduction In the calculation of frictional force of a flexible element such as a belt, rope or cable wrapped around the cylinder, the famous Euler's belt formula (Hashimoto, 2006) or simply known as the belt friction equation (Joseph F. Shelley, 1990) is used. The formula is useful for designing a belt drive or band brake (J. A. Williams, 1994). On the other hand, a belt or rope is conveniently used to tighten a luggage to a carrier or lift up the luggage from the carrier. In that case, for the sake of adjusting the belt length and keeping an appropriate tension during transportation, various kinds of belt buckles are used. These belt buckles have been devised empirically and there was no theory about why it can fix the belt. The first purpose of this chapter is to present the theory of belt buckle clearly by considering the self-locking mechanism generated by wrapping the belt on the belt. Making use of the belt tension for a locking mechanism, a belt buckle with no locking mechanism can be made. The principle and some basic property of this new belt buckle are also shown. 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 F ij = F ji 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 N ij = N ji Normal force of belt between point P i and P j , N P i Boundary of contact angle R Radius of main cylinder, m New Tribological Ways 236 T i 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 θ Angle θ i Angle of point P i θ ij =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. T 1 and T 4 (T 1 >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 T 1 , T 2 and T 3 (Joseph F. Shelley, 1990). 12 34 1223 , b Te T TeT μθ μθ == (1) T 4 ’ and T 4 ” are of inner belt tension at P 1 and P 2 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 dN e T d μθθ θ − = (2) Making use of T 4 ’ and T 4 ”, 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 () 25 4 () 12 4 () 16 4 " ' dN e T d dN e T d dN e T d μθ θ μθ θ μθ θ θ θ θ − − − ⎫ = ⎪ ⎪ = ⎬ ⎪ = ⎪ ⎭ (3) The frictional force between P 1 and P 6 is 6 16 1 16 16 4 (1)FdNeT θ μθ θ μ ==− ∫ (4) The inner belt tension T 4 ’ is the sum of the frictional force F 16 and the belt tension T 4 . 16 4416 4 'TTFeT μθ =+ = (5) The frictional force F 12 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 and dN 12 . The normal force dN b is exerted from the outer belt. The other normal force dN 12 is generated by the inner belt tension. So, F 12in is given by 11 12 12 22 2 12 12 4 (1)(1)' b in b b T FdNdNe eT θθ μθ μθ θθ μ μμ μ =+ =−+− ∫∫ (6) Making use of Eq. (2), the frictional force F 12out acting on the belt-belt boundary can be written as 1 12 2 12 2 (1) b out b b FdNeT θ μθ θ μ ==− ∫ (7) The frictional force F 12 is the sum of Eqs. (6) and (7). 12 12 12 2 4 (1)1 (1)' b b Fe Te T μθ μθ μ μ ⎛⎞ =−++− ⎜⎟ ⎝⎠ (8) As the belt tension T 4 ” is the sum of F 12 and T 4 ’ , making use of Eq. (5) and (8), T 4 ” can be written as 26 12 4124 4 2 "' (1)1 b b TFTeTe T μθ μ θ μ μ ⎛⎞ =+= + − + ⎜⎟ ⎝⎠ (9) Making use of Eq. (3), the frictional force F 25 can be written as 2 25 5 25 25 4 (1)"FdNeT θ μθ θ μ ==− ∫ (10) As the belt tension T 3 is the sum of F 25 and T 4 ”, making use of Eqs. (9) and (10), T 3 can be expressed as 25 26 12 3254 4 2 "(1)1 b b TF T e eT e T μθ μθ μ θ μ μ ⎧ ⎫ ⎛⎞ ⎪ ⎪ =+= + − + ⎜⎟ ⎨ ⎬ ⎪ ⎪ ⎝⎠ ⎩⎭ (11) [...]... Property of Flexible Element Substituting Eq (84 ) into ( 78) gives T1 = e ( μθ 3 1 − μ θ 3 − θ 1 + 2θ 1e μ (θ 3 −θ 1 ) ) T4 (85 ) In order to consider the smallest wrapping angle of three times wrapping, substituting θ1=0 into Eq (85 ) gives, T1 = e μθ 3 T4 1 − μθ 3 (86 ) On the other hand, substituting θ1=θ3 into Eq (85 ) gives, T1 = e μθ 3 T4 1 − 2 μθ 3 (87 ) Equation (87 ) shows the relation of belt tension with... +θ 3 ) , B=− (e μb θ 3 + 1)( e μbθ1 − 1) e μb (θ1 +θ3 ) (80 ) then A− B = e e μb (θ1 +θ 3 ) μb (θ1 +θ 3 ) =1 (81 ) Substituting Eq (81 ) and μ=0 into Eq ( 78) gives T1=T4 In the case of μb=0, limiting operations are required For the term A in Eq (79), lim μ μθ μθ e −e = μ (θ 3 − θ 1 ) μb (82 ) lim μ μθ μθ e +1 e − 1 = 2μθ 1 μb ) (83 ) , B = 2μθ 1 (84 ) μb → 0 ( b 3 b 1 ) For the term B in Eq (79), μb → 0... is satisfied when the denominator of Eqs (86 ) and (87 ) become 0, so that in the case of θ1=θ3, only 1/2 of the coefficient of friction is required for self locking in compared with the case of θ1=0 In the case of μb=μ, Eq (79) becomes, 1 A= e 2μ θ 3 , B=0 (88 ) so that Eq ( 78) becomes, μθ T1 = e 1 μ (θ + 2θ ) T4 = e 1 3 T4 A− B (89 ) Substituting θ1=0 into Eq (89 ) gives, T1 = e 2 μθ 3 T4 (90) T1 = e 3... 1 + 4n 2π 2 − 4nπ t + t 2 3 ⎫2 ⎧ t ⎞ 2⎪ ⎪ ⎛ 2nπ ⎨ a2 + ⎜ 1 − ⎟ b ⎬ 2 nπ ⎠ ⎝ ⎪ ⎪ ⎩ ⎭ 2 ), 2 t ⎞ 2 ⎛ ds = a 2 + ⎜ 1 − ⎟ b dt 2nπ ⎠ ⎝ (1 18) 262 New Tribological Ways Fig 23 Change of tension ratio T2/T1 with coefficient ratio b/a of spiral Substituting Eq (1 18) into Eq ( 98) and integrating with respect to parameter t from t1=0 to t2=2nπ gives ⎧ 1 ⎛ ⎛T ⎞ 1 + β 2γ 2 ⎪ log ⎜ 2 ⎟ = μ ⎨ log ⎜ ⎜ 1 + β 2γ 2 +... (Joseph F Shelley, 1990), analysis starts with the conventional equation T1 = e μbθ 1 T2 = e μbθ 1 T3 ( 28) Frictional Property of Flexible Element 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 243 244 New Tribological Ways Fig 9 Mechanical model of belt wrapped two times around an axis The belt tension T2 or T3 can be expressed... θ1 for self-locking with ratio of the coefficients of friction κ Fig 11 Fraction of belt tension T4 /T1 decreases rapidly with increment of over-wrapping angle θ1 for the case of smaller κ 2 48 New Tribological Ways Fig 12 Polyethylene film was wrapped together with belt to reduce the coefficient of friction μb Self-locking was recognized in experiment with polyethylene film But it never occurred without... into the left hand side of Eq (74) gives, μθ1 T4 (76) 252 New Tribological Ways ( ⎛ μ ⎞ e T1 " = ⎜ − 1 ⎟ ⎝ μb ⎠ ( 1− e = μbθ 3 e )( μbθ 3 +1 e e −e −1 μb (θ 1 +θ 3 ) )T + e μθ 1 1 T4 = BT1 + e μθ1 T4 (77) ⎛ μ ⎞ ⎜ − 1⎟ μb ⎝ ⎠ T = AT 1 1 + μ θ −θ ) μbθ 1 μb (θ 1 +θ 3 ) μbθ 1 ( 3 1 ) Equation (77) can be written in the form of T1 = e μθ1 T4 A−B ( 78) where ( 1− e A= μbθ 3 −e μ bθ 1 ⎛ μ ⎞ ) ⎝ μ − 1⎠ ⎜ ⎟ b... 10 1 20 ζ x μb =0.15 μb =0.1 40 60 80 100 T4 T1 μ =0 120 140 160 Unfolding angle of buckle ζ, deg 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 μb especially in the vicinity of locking condition It is very sensitive to angle ζ 240 New Tribological Ways Ratio of belt tension T1/T4 100... directional change in arm 242 New Tribological Ways 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 μb Making use of Eqs (13) and (24), the fraction of belt tension can be calculated Figure 8 shows some results The...2 38 New Tribological Ways Substituting Eq (1) into Eq (11) to eliminate T2 gives T3 = e μθ56 1−e μ (θ 34 +θ 25 ) (e μb θ 12 − 1) ( 1 + μ / μb ) T4 (12) Substituting Eq (1) into Eq (12) to get the relation between . ISSN: 1 083 -7450. Fell, J.T. & Newton J.M. (1970). Determination of Tablet Strength by Diametrical- Compression Test . International Journal of Pharmaceutics, 59, 688 -691, ISSN: 03 78- 5173 03 78- 5173. Yusof, Y. A., Smith, A. C., & Briscoe, B. J. (2005). Roll compaction of maize powder. Chemical Engineering Science , 60 (14), 3919-3931, ISSN 0009-2509. New Tribological Ways. 2 "(1)1 b b TF T e eT e T μθ μθ μ θ μ μ ⎧ ⎫ ⎛⎞ ⎪ ⎪ =+= + − + ⎜⎟ ⎨ ⎬ ⎪ ⎪ ⎝⎠ ⎩⎭ (11) New Tribological Ways 2 38 Substituting Eq. (1) into Eq. (11) to eliminate T 2 gives () 56 12 34 25 34 () 1(1)1/ b b e TT ee μ θ μθ μθ