ON THE FRICTION OF THE THREAD WITH RIGIDITY

Abstract

The frictional behaviours of the thread wrapped around cylinder have been studied with a consideration on the flexual rigidity of thread. Firstly, assuming that the thread is a completely elastic body and the coefficient of friction between the thread and cylinder is constant, the studies were extended to general thread.1). Basic Equation (Fig. 2)where, T0, T1; thread tension acting at the contact end, F0 F1; reactive force acting at the contact end against cylinder, μ; coefficient of friction between thread and cylider, θ; contact angle of thread wrapping a round the cylinder.2). General case (Fig. 3)Using equation (a), the following is obtained.where, t0, t1; thread tension acting at the thread end, ƒ0, ƒ1; normal force acting at the thread end, θ0, θ1; contact angle to the standard line, α0, α1; angle between thread at its end and perpendicular to the standard line.3). When the thread is long enough (Fig. 5)Using the relationships between the angles and equation (b) following formulas are obtained: The relationships between t0 and t1 are;where, E; Young's modulus of the thread, I; moment of inertia of area of the thread, i.e. EI shows the flexual rigidity of thread, ρ; radius of cylinder.the relationship between θ0 and θ1 is:But, where the point contact or θ0+θ1=0, and4). It has been found on the constant coefficient to friction as following:i) The frictional force with rigidity is smaller on the thread than without (Fig. 8 and Fig. 9).ii) The frictional behaviour may be approached to condition without rigidity as thread tension becomes greater (Fig. 8 and Fig. 9).iii) The greater the coefficient of friction and turning angle are the more the frictions are influenced by flexual rigidity. And as these values are smaller these influences are trivial (Fig. 10).iv) When the contact is at point, the ratio of final tension to initial tension is not changed by the tension of the thread as shown in equation (e).v) When the contact is at point, its position does not change with the thread tension, as shown in equation (f.)5). In general threads similar tendencies may be expected (Fig. 11).