Sommario
Il lavoro riferisce i risultati di una serie di prove sistematiche effettuate con una apparecchiatura sperimentale appositamente costruita, e intese a verificare le condizioni di stabilità delle oscillazioni torsionali autoeccitate in trasmissioni omo cinetiche a doppio giunto di Cardano.
Per un angolo di inclinazione di 30° con diversi valori dello smorzamento e facendo variare la velocità di rotazione entro una gamma comprendente i valori critici corrispondenti alla risonanza con la pulsazione naturale del sistema e alle sure frazioni 1/2 e 1/3, i rilievi sperimentali appaiono in buon accordo con le previsioni teoriche, in particolare con quelle della teoria non-lineare.
Essi rivelano in effetti ampiezze di vibrazione molto più grandi del normale in prossimità dei valori critici. Tuttavia esistono interessanti divergenze: i fenomeni di jump e di isteresi sulle curve di risposta si presentano con modalità diverse da quelle previste teoricamente e lo smorzamento sembra avere in realtà un'efficacia molto inferiore al previsto nel ridurre le ampiezze delle oscillazioni e l'estensione delle zone critiche.
Summary
The paper reports experimental results from a series of systematic tests carried out by means of a specially built testing apparatus, and aimed at verifying stability conditions for self-excited torsional oscillations in constant-velocity two-Cardan-joint drives. For a 30° misalignement, with different dampings and varying the revolving speed over a range taking in the critical values corresponding to resonance with the system's natural frequency and to its 1/2 and 1/3 fractions, the experimental findings appear in good agreement with theoretical predictions, expecially with the ones from non-linear theory.
In fact they show steady almost periodic oscillations having amplitude by far higher than normal in the vicinity of critical speed values. There are however some noteworthy discrepancies: jump and hysteresis phenomena on the response curves turn out to have different peculiarities from those theoretically predicted and damping appears to have a much smaller effectiveness than predicted in reducing oscillation amplitudes and critical regions extension.
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Abbreviations
- α :
-
Misalignement angle of the intermediate shaft with reference to the end shafts (parallel to each other)
- ω:
-
Revolving speed of the shafting
- \(\left. \begin{gathered} \phi _1 = \omega t \hfill \\ \phi _2 = \phi \hfill \\ \end{gathered} \right\}\) :
-
Input and output shaft rotation angles
- \(\theta = \phi _2 - \phi _1 = \phi - \omega t\) :
-
Relative motion of the output shaft with reference to the input one
- k :
-
Torsional stiffness of the intermediate shaft
- h :
-
Torsional stiffness of the shafting (misaligned and rotated) i.e. restoring torque coefficient on the output shaft
- J :
-
Mass moment of inertia of the flywheel-like masses rotating with output shaft
- \(\Omega _n = \sqrt {\frac{k}{J}} \) :
-
Natural circular frequency of the system
- \(v_n = \sqrt {\frac{{\Omega _n }}{{2\pi }}} \) :
-
Natural frequency of the system
- \(\eta = \frac{\omega }{{\Omega _n }}\) :
-
Typical ratio of the revolving speed
- s :
-
Coefficient of torsional viscous resistance on the output shaft
- \(\gamma = \frac{s}{{2J\Omega _n }}\) :
-
Viscous damping coefficient of the system
- \(\Gamma = \ln \frac{{\theta _m }}{{\theta _{m + 1} }}\) :
-
Logaritmic decrement of the damped oscillations
- i :
-
Excitation current of the damping device.
Bibliografy
A. Costa:Torsional Oscillations in constant-velocity drives incorporating two Cardan joints: the present state of knowledge and the design of an experimental research, in «Meccanica», n. 4, 1982.
A. Dornig:Instabilità dinamiche nelle macchine dovute alla presenza di trasmissioni a doppio giunto cardanico, In «Rivista di Ingegneria», n. 7, luglio 1965.
B. Porter, R.W. Gregory:Nonlinear torsional oscillations of system incorporating a Hooke's joint, in «Journ. Mech. Eng. Sc.», 1963, n. 2.
N.W. Mc Lachlan:Ordinary non-linear differential equations in engineering and physical science, Oxford, Clarendon Press, 1950.
W.J. Cunningham:Introduction to non-linear analysis, McGraw Hill, 1958.
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Costa, A., Davoli, P. An experimental investigation on torsional oscillations in constant velocity two-Cardan-joint drives. Meccanica 18, 34–53 (1983). https://doi.org/10.1007/BF02156099
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DOI: https://doi.org/10.1007/BF02156099