On the Shape Factor for the Maximum Pressure. of the Undercut Groove Form

Transcription

1 Int. Journal of Math. Analysis, Vol. 1, 007, no. 16, On the Shape Factor for the Maximum Pressure of the Undercut Groove Form C. E. İmrak and İ. Gerdemeli Istanbul Technical University Faculty of Mechanical Engineering Mech. Eng. Dept., Gumussuyu TR-34437, Istanbul, Turkey Abstract The effective coefficient of friction between the ropes and grooves depends on the geometrical shapes of the grooves and the actual coefficients of friction. The effect of the changes in the groove angle and undercutting angle on the coefficient of friction and the traction are also studied. Therefore this work introduces the concept of shape factors for the maximum specific pressure for undercut grooves namely round and undercut grooves and also derives it by means of stress fuction method. Mathematics Subject Classification: 74A10, 74A55, 33C10 Keywords : Coefficient of friction, stress function, shape factor, undercut groove. 1 Introduction The effective coefficient of friction between the ropes and grooves depends on the geometrical shapes of the grooves and the actual coefficients of friction. The groove shapes of traction sheaves are designed to produce pinch the steel wire rope. The driving sheaves are widely employed for power transmission to the

2 784 C. E. İmrak and İ. Gerdemeli ropes driving elevators, cable cars, funiculars, etc. The maximum traction that can be developed in the sheave groove is a function of the actual coefficient of friction between the rope and the groove and the angle of contact that the rope makes with the circumference of the sheave. The groove form favorably increases the effective coefficient of friction between the rope and the groove at the expense of increasing the pressure and wear on the groove surface. The trade off between the traction produced and pressure causing abrasion of the groove may be best explained by the concept of the shape factors for both the coefficient of friction and the maximum specific pressure for undercut grooves. Thus, the groove form effects the magnitude of the traction force on driving sheaves for power transmission. The problem of calculating the coefficient of friction and the specific pressure between the rope and the groove is of great importance and many papers have been devoted to this subject [1-4]. To obtain the pressure distribution on the contact surface of undercut grooves, the Airy s stress function method was proposed by Imrak [3]. The effect of the groove geometry and the angle of wrap on traction were investigated by Imrak and Ozkirim [4]. They tabulated the ratio of the forces for different values of the angle of wrap. There are many of published studies on stress function application in solid mechanics [1-10]. To obtain the pressure distribution on the contact surface of undercut groves, the Airy s stress function method appeared in the paper by Imrak [8]. The effect of the groove geometry and the angle of wrap on the traction were investigated and tabulated the ratio of the forces for different angle values by Imrak and Ozkirim [9]. The contact area between the rope and the groove is smaller with the undercut grooves than the round grooves since the rope looses the contact with the groove where the undercut is machined. Thus an undercut groove provides a tighter gripping action due to an increased groove pressure and their traction capability is greater than that of a round groove. However a round groove has longer rope life and a lower level of noise because of lower groove pressure at high speeds. When the problem of insufficient traction arise with the round groove it is should be noted that it may be overcome by increasing the angle of wrap, by changing the groove form into an undercut groove with an adequate shape or by using a material with a higher coefficient of friction. In this paper the expression for the pressure distribution on the contact surface of round and/or undercut groves is obtained by means of stress function method, the shape factors for the maximum specific pressure and the coefficient of friction are derived and presented. In this paper the application of Stress Function technique for determining the shape factors for the maximum specific pressure for two types of undercut grooves, namely round and undercut grooves are presented. The effect of the changes in the groove angle and undercutting angle on the coefficient of friction and the traction are also studied.

3 Shape factor for maximum pressure 785 Basic Equations Airy introduced his stress function as a device for solving certain problems in linear elastostatics for homogeneous isotropic bodies. Due to the normal force and symmetric loading in the traction drive, plane stress is only employed. The equation of compability, which mans that the body must be physically pieced together in terms of stress function is [11] φ φ φ φ = + + = 0 r r r r r r r r (1) ϕ ϕ where is the Laplacian Operator. The stress function is developed and used to solve classic two-dimensional problems fundamental to stress analysis. The Airy Stress Function approach works best for problems where a solid is subjected to prescribed tractions on its boundary, rather than prescribed displacements [8]. By symmetric straining, one can find / r = d / dr. Therefore Eq.(1) becomes d 1 d d φ 1 dφ + + = 0. () dr r dr dr r dr Thus the solution reduces to finding a solution of the differential equation of compatibility which satisfies the boundary conditions of the problem. To obtain a solution of these two equations, one can write an arbitrary function φ = φ(r,ϕ ). This arbitrary function is called the Stress Function [1]. In the case of plane stress and in the event that the body forces are negligible the differential equations of equilibrium in polar coordinates are as follows φ φ φ σ r = ; σ t = ; τ = - (3) t r r t and the boundary conditions are σ = r p, σ = 0 t, and τ = 0. dt = r sin dϕ r dϕ with respect for radial derivative. One can assume the stress functionv(r, ϕ ) = cη r ϕ sinϕ, where c is a constant. It can be easily verified that the stress function satisfies the equation of compability. Thus it represents the true stress function. Then Eqs. (3) can be written in the following forms: 1 V 1 V cη σ = + cosϕ r r ϕ r r = V, σ ϕ = = 0, r r 1 V 1 V 1 V τ = = = 0. (4) r r ϕ r ϕ r r ϕ where V is Airy Stress Function, η is the radial force distribution, c is the constant.

4 786 C. E. İmrak and İ. Gerdemeli 3 The Shape Factor for the Maximum Pressure The shape factor means that when maximum rope tension, the pitch diameter of the sheave and the rope diameter are fixed as rated parameters the maximum pressure can be adjusted, to some, extend by choosing the proper values for the angle of the undercutting β and the angle of the outer normal lines of the contact area δ. Therefore the maximum pressure can be reduced to a level that the wear of the rope and the sheave groove can minimize and the life of them may be extended. The geometry and the loading of an undercut groove are illustrated in Fig. 1. The angle of groove γ can reach the maximum value of 180. The angle of the outer normal lines of the contact area δ may have the maximum value of 180. The angle of the undercutting β must not be greater than 105 and γ + δ = 360 [15]. Figure 1: The undercut groove form. The boundary conditions along the area of contact, for r = r 1 = d, in the radial plane at an angle of α are expressed by D dn = η dα = σ r cosϕ da, (5) β / where da represents an infinitesimal contact area with the dimensions( D / )dα along the arc of the wrap of the rope on the sheave and ( d / )dϕ in the radial plane of the sheave, where d is the rope diameter, hence da = d D dϕ dα / 4. It can easily be proved that the first two of the boundary conditions are satisfied by the last two of the equations of the distribution of stress. Inserting da and Eq. (5) into the Eq. (4), one obtains D η dα = D dα cη and then Eq.(5) takes the form δ / δ / β / cos ϕ dϕ, (5)

5 Shape factor for maximum pressure 787 δ 1 = 4 c cos ϕ dϕ, (6) β the integration gives 1 c =. (7) δ - β + sin δ - sin β Inserting the constant c given by Eq.(7) and the radial force distribution η per unit length along the circumference of the sheave given by Eq. (7) into the Eq. (4), one obtains 4S cosϕ σ r =, (8) D( δ - β + sin δ - sin β ) r where S is the rope tension at the point on the arc of the rope with an angleα, β is the angle of the undercutting and δ is the angle of the outer normal lines of the contact area. Therefore the distribution of specific pressure along the area of contact is 8S cosϕ p = σ r =. (9) Dd( δ - β + sin δ - sin β ) where r = r 1 = d. The maximum specific pressure exists at the edge of the undercut where δ = β / and at the location where the rope tension is maximum value. Therefore the maximum pressure is given by the expression 8 S max cosβ / p max =. (10) D d δ - β + sin δ sin β where S max is the maximum rope tension, D is the pitch diameter of the sheave, d is the rope diameter, β is the angle of the undercutting and δ is the angle of the outer normal lines of the contact area. The groove shape factor for the maximum pressure can be expressed by 8 cosβ / f =. (11) δ - β + sinδ sin β The variation of the groove shape factor for the maximum pressure respect to the angle of the outer normal lines of the contact area for various values of the angle of the undercutting β is illustrated in Fig..

6 788 C. E. İmrak and İ. Gerdemeli Figure : The variation of the groove shape factor for the maximum pressure for various values of the angle of the undercutting. It is clearly seen from Fig. that when the angle of groove decreases the traction improves, but also the specific pressure and resultant wear of both the grooves and ropes. The round groove gives lower traction but a longer rope life, lower specific pressure and lower degree of noise than the undercut. When it is essential to use the round grooves the traction capability can still be improved by using non-metallic groove liners with a high coefficient of friction. It is also advised that the angle of the undercutting should be preferably under 90 and must not be greater than 105 whenever the undercut grooves are in use. 4 Conclusions In this paper the expression for the pressure distribution on the contact surface of round and/or undercut groves is obtained by means of stress function method, the shape factors for the maximum specific pressure and the coefficient of friction are derived and presented. In this paper the application of Stress Function technique for determining the shape factors for the maximum specific pressure for two types of undercut grooves, namely round and undercut grooves are presented. The effect of the changes in the groove angle and undercutting angle on the coefficient of friction and the traction are also studied.

7 Shape factor for maximum pressure 789 The groove form favorably increases the effective coefficient of friction between the rope and the groove since the radial force due to the rope tension produces greater normal and friction forces acting along the area of contact given by the shape of the groove. Therefore this work introduces the concept of shape factors for both the coefficient of friction and the maximum specific pressure for undercut grooves and also derives them by means of Airy s stress function method. Acknowledgement It is pleasure to acknowledge many stimulating correspondences with Professor E. Minchev. References [1] D. E. Jesson and W. N. Cottingham, Investigation of beam theory using the Airy stress function: Coupled with analytic function theory, Journal of Engineering Mathematics, 0 (1986), [] R.A. Bourgois, Application of the generalized Airy stress function to problems on elastic vibrations of hollow cylinders, Journal of Applied Mechanics, Transactions ASME, 40 (1973), [3] C.F. Yen, New membrane element based on Airy s stress function, Journal Institusi Jurutera Malaysia, 41 (1987), [4] A.M. Tarantino, Thin hyperelastic sheets of compressible material: Field equations, Airy stress function and an application in fracture mechanics, Journal of Elasticity, 44 (1996), [5] E.E. Gdoutos, Stress function interface and boundary conditions in anisotropic materials, Journal of Applied Mechanics, Transactions ASME, 49 (198), [6] H.A. Sosa and L.Y. Bahar, Transition from airy stress function to state space formulation of elasticity, Journal of the Franklin Institute, 39 (199), 817. [7] H. Hasegawa, On the strain functions and complex stress functions in the two dimensional theory of elasticity, Nippon Kikai Gakkai Ronbunshu, A Hen, 53 (1987), [8] C.E. Imrak, Verleich des Rillenfaktors und des Reibungsbeiwertes der Treibscheiben-Seilrille, Lift-Report, (1998), [9] C.E. Imrak and M. Ozkirim, Calculating the pressure distribution in undercut grooves: stress-function solution, Proceedings of Elevcon 00, Milan,

8 790 C. E. İmrak and İ. Gerdemeli [10] R. Fosdick, Generalized Airy stress function, Meccanica, 38 (003), [11] S. Timoshenko and J.N. Goodier, Theory of Elasticity, McGraw-Hill Book Co. New York, [1] A.C. Ugural and S.K. Fenster, Advanced Strength and Applied Elasticity, Prentice Hall, New Jersey, [13] J. Hannah and R.C. Stephens, Mechanics of Machines, Edward Arnold Publ., London, [14] O.T. Bruhns, Advanced Mechanics of Solids, Springer Verlag, Berlin, 003. [15] L. Janovsky, Elevator Mechanical Design, Ellis Horwood Ltd., New York, Received: February 3, 007