DEPENDENCE OF THE CAM FOLLOWER LOADING ON THE SYSTEM ELASTICITY
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1 SCIENTIFIC BULLETIN OF LODZ TECHNICAL UNIVERSITY Nr 89, Textiles 60, 00 ZESZYTY NAUKOWE POLITECHNIKI ŁÓDZKIEJ Nr 89, WŁÓKIENNICTWO z 60, 00 Pages: JERZY ZAJACZKOWSKI Loz Technical University Loz, Zwirki 36, Loz, Polan DEPENDENCE OF THE CAM FOLLOWER LOADING ON THE SYSTEM ELASTICITY Reviewer: Janusz Lipinski, PhD, DSc The paper is concerne with the problem of epenence of the loaing of the cam mechanism on the elasticity of its elements The mechanisms with rigi elements, with spring mass system, with an elastic cam shaft an with the elastic rocker shaft are stuie The motion of the mechanism is escribe by nonlinear ifferential equations The moment acting on the rocker is calculate an plotte INTRODUCTION The cam mechanisms are wiely use in the textile machinery Vibrations of shafts couple by mechanisms have been stuie in work [] The phenomenon of torsional buckling of shafts has been escribe in work [] In paper [3] it has been shown that the vibrations are mainly ue the collision of the rotary an oscillatory motion of elements of the mechanism It was pointe out that the elasticity of the riving belt improves the system ynamics The purpose of this paper is to stuy the effect of the cam mechanism elasticity on the rocker loaing EQUATIONS OF MOTION
2 6 J Zajaczkowski The cam mechanisms consiere are shown in Figure They are compose of the rotating groove cams an the oscillating followers The mechanisms with rigi elements (a), with a spring an a mass (b), with an elastic cam shaft (c) an an elastic follower shaft () are consiere Figure The cam mechanisms with:(a) rigi elements, (b) spring an mass, ( c ) elastic cam shaft, ( ) elastic follower shaft The equation of the motion of the cam mechanism with rigi elements, shown in Figure a, has the form A B B The moment of the forces acting on the follower is foun to be M r M m (a)
3 Cam follower loaing 7 M B M B B r Mr (a) The motion of the cam mechanism with spring an oscillating mass, shown in Figure b, is governe be the equations x m x c kx m e x 0, A m e x B me x x me x B M 0 The moment of the forces acting on the follower is m (b) x M B e x me x B (b) The equations of the motion of the cam mechanism with an elastic cam shaft, shown in Figure c, can be written in the following form A D s0 M m 0, A B B
4 8 J Zajaczkowski D 0 0 s 0 The moment of the forces acting on the follower is given by (c) M B B (c) The approximate natural frequency is =(s/a 0 +s/a ) The equations of the motion of the cam mechanism with the elastic follower shaft, shown in Figure, have the form A B B H k M m 0, B H k The moment of the forces acting on the follower is 0 () H k M B B The approximate value of the natural frequency is =(k/b ) () The motor torque is given be the following equation
5 Cam follower loaing 9 Mm T C M m (3) The ifferential equations () escribing the motion of the cam mechanisms can be integrate numerically together with the equation of the motor torque (3) an the moment () of the forces acting on the follower can be calculate RESULTS AND DISCUSSION The function relating the motion of the follower to the motion of the cam was taken in the form S cos 0 0 The calculations were carrie out for =60/80, S =30/80, =000/30, T=/, C=0 It was assume that at the initial moment the angular spee of the motor was equal to, the motor torque M m (0)=0, the eformations of the elastic elements were assume to be equal to zero The total inertia of the rocker was kept unchange for all mechanisms The torques for mechanisms (Figure a,b,c,) are shown in Figures a,b,c, for (a) A=00, B=00, (b) A=00, B=00, e=0, m=, k= m, c=0(mk), A 0 =00, A =00, B =00, s=(5 /(/A 0 +/A ), D=0(s/(/A 0 +/A )), () A =00, B =B =00, k=(5 B, H=0(B k), respectively Comparing the results shown in Figure a,b,c an, one may see that the elasticity of the follower shaft has significant influence on the system behaviour
6 0 J Zajaczkowski Figure The moment of the forces acting on the cam follower for cam mechanisms with: (a) rigi elements, (b) spring an mass, ( c ) elastic cam shaft, ( ) elastic follower shaft REFERENCES Zajaczkowski J: Torsional buckling of shafts couple by mechanisms Journal of Soun an Vibration 73(4), (994) Zajaczkowski J: Vibrations of shafts couple by mechanisms Journal of Soun an Vibration 77(5), (994) 3 Zajaczkowski J: Torsional vibration of a camshaft Zeszyty Naukowe Politechniki ózkiej Nr 736, Wókiennictwo, z 53, 99-03, (996)
7 Cam follower loaing ZALEŻNOŚĆ OBCIĄŻENIA POPYCHACZA MECHANIZMU KRZYWKOWEGO OD SPRĘŻYSTOŚCI UKŁADU Streszczenie Praca otyczy zaganienia zależności obciążenia mechanizmu krzywkowego o sprężystości jego elementów Baane smechanizmy z elementami sztywnymi, ze sprężystym wałem krzywkowym, ze sprężystym wałem popychacza oraz z ukłaem rgającym złożonym z masy i sprężyny Ruch mechanizmu opisany jest za pomoc nieliniowych równa różniczkowych Moment ziałający na popychacz, wyznaczony po rozwiązaniu równa różniczkowych, jest przestawiony wykreślnie
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