RESONANCE OF TORSIONAL VIBRATION OF SHAFTS COUPLED BY MECHANISMS

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Radosław Orzechowski
7 lat temu
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1 SCIENTIFIC BULLETIN OF LOZ TECHNICAL UNIVERSITY Nr 78, TEXTILES 55, 997 ZESZYTY NAUKOWE POLITECHNIKI ŁÓZKIEJ Nr 78, WŁÓKIENNICTWO z. 55, 997 Pages: 8- JERZY ZAJACZKOWSKI Loz Technical University Zwirki 6, Loz, Polan RESONANCE OF TORSIONAL VIBRATION OF SHAFTS COUPLE BY MECHANISMS Reviever: prof. r hab. Krzysztof ems Receive The paper is concerne with the resonance of torsional vibrations of two shafts couple by two mechanisms. The motion of the system is escribe by a set of non-linear ifferential equations. The ynamic motor characteristic is taken into account. The attractors in the phase space associate with moal co-orinates are foun.. INTROUCTION The problem of the torsional vibration of shafts couple by mechanisms has been investigate in paper []. The phenomenon of the ynamic torsional buckling an the post-buckling irregular motion have been stuie in references [,]. The catastrophic bifurcations of the perioic attractors have been foun in paper [4]. The intermittent motion of the system has been stuie in papers [5,6]. The analysis to follow focuses on a resonance behaviour of the non-symmetric system.

2 6 Jerzy Zajączkowski. EQUATIONS OF MOTION The system consiere, shown in Figure, consists of a motor, two cam mechanisms an two isks attache to the cam shaft. It is assume that the ynamic characteristic of the motor is given by [7] T M t A where T, C, are the motor constants. M A C, t () Figure. Shafts couple by two cam mechanisms. The equations of the motion of the system may be written in the form A s M A t t t, A B B t t H t t t t t t s s k, A t t t t t

3 Resonance of torsional vibrations... 7 A B s s, B t t t t H s k. t t () Here,, are the rotation angle of shafts, A, B are the mass moments of inertia, s, k are the torsional stiffnesses of shafts,, H are the coefficients of viscous amping. The function is assume in the form cos A ; The amping coefficients are taken as () A A A A s A A A A s, A A A A s H B B B B k,. (4) The natural vibrations of the system having average inertia an stiffness are escribe by the following equation A A B A t A B where s s s s s k s k s s s s k s s k. (5)

4 8 Jerzy Zajączkowski Equation (5) can be rewritten in a matrix form (6) A S. t The natural frequencies n an the eigenvectors Y n are foun from the following equation S A Y. n The eigenvectors Y n of equation (8) are orthonormalize with respect to the matrix A an assemble into a single square matrix T T n Y Y, Y, Y, Y, Y AY I, Y SY iag,.. n (9) Here, I is the unit matrix. The first element of each vector of the matrix Y is taken to be positive. The moal co-orinates for consiere system are foun in the form u py v py,. t () The parameter p can be foun from the relationship between the co-orinates for the rigi system (7) (8) py Transformation () is applie to the numerically foun solution of equations (,) (). NUMERICAL RESULTS AN ISCUSSION Numerical calculations were carrie out for A =4 /8ra, the amping coefficient =, the motor constants C=.6Nms/ra, T /, the moments of inertia A =.4, A =., A =.4, A =., B =5, B kgm, the stiffnesses s s s =, k =(Nm/ra). The natural frequencies were foun to be =, =5.5, =89.7, =.59ra/s, the moal matrix was foun as

5 Resonance of torsional vibrations Y The set of equations (,) was being numerically integrate until the orbits, subject to transformation (), became close curves. The projections of those orbits on the phase planes in the parameter space of are shown in Figure. The orbits represent the torsional vibrations of the shaft. The vibrations result from the collision of the tenency to the constant spee motion of the isks an the tenency to the fluctuating spee motion of the cams. It can be seen from the Figure that for small stiffness of the motor characteristic the system cannot pass the resonance an the increase in results in the increase of the amplitue of torsional vibrations.

6 Jerzy Zajączkowski Figure. Projections of trajectories of the motion on the phase planes in the parameter space of REFERENCES [] Zajaczkowski J.: Torsional vibrations of shafts couple by mechanisms. Journal of Soun an Vibration. 6(), -7 (987)

7 Resonance of torsional vibrations... [] Zajaczkowski J.: Torsional buckling of shafts couple by mechanisms. Journal of Soun an Vibration. 7(4), (994). [] Zajaczkowski J.: Lyapunov stability of shafts couple by mechanisms. Zeszyty Naukowe Politechniki Łózkiej Nr 76 (996). Włókiennictwo z. 5, 5- [4] Zajaczkowski J.: Vibrations of shafts couple by mechanisms. Journal of Soun an Vibration. 77(5), 79-7 (994) [5] Zajaczkowski J.: Increase of stresses ue to intermittent motion of a cam mechanism. Zeszyty Naukowe Politechniki Łózkiej Nr 7 (995). Włókiennictwo z. 5, 7- (995). [6] Zajaczkowski J.: Torsional vibration of camshaft. Zeszyty Naukowe Politechniki Łózkiej Nr 76 (996). Włókiennictwo z. 5, 9-. [7] Veitz V.L., Martynenko. M.: Mechanism an Machine Theory 8, 97, -5. Frequency characteristic of a machine assembly with a self-locking mechanism. REZONANS RGAŃ SKRĘTNYCH WAŁÓW POŁĄCZONYCH MECHANIZMAMI Streszczenie W pracy baano rgania rezonansowe ukłau wałów połączonych mechanizmami. Ruch mechanizmu opisano za pomocą ukłau nieliniowch równań różniczkowych. Uwzglęniono ynamiczną charakterystykę silnika. Znaleziono atraktory w przestrzeni fazowej związanej ze współrzęnymi moalnymi. Łóź, Politechnika Łózka Instytut Maszyn i Urzązeń Włókienniczych

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