Abstract
Self-excited oscillations are of considerable importance in mechanical systems, and are capable of completely defeating the purpose for which a device is intended, including the possibility of self-destruction. The results of a theoretical and experimental investigation of this phenomenon are presented, with particular reference to the effect of variations in the system parameters upon the critical threshold velocity at which such oscillations commence. Good correspondence between theory and experiment is observed. Results are presented in terms of dimensionless parameters β (dependent upon system stiffness and solid friction) and ζ (internal damping ratio). It is concluded that for β<0.0001 (very stiff system) there is little danger of self-excited oscillations, and also that the critical threshold velocity is extremely sensitive to small changes in ζ when ζ is small (on the order of ζ=0.01).
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Abbreviations
- c :
-
damping coefficient, lb-in.-sec
- d :
-
diameter of drive shaft, in.
- E s :
-
shear modulus=12,000,000 for steel, lb/in.2
- f 1 :
-
frictional torque associated with kinetic coefficient of friction,ü k , lb-in.
- f 2 :
-
frictional torque associated with static coefficient of friction,ü s , lb-in.
- I :
-
polar moment of inertia of rotor, lb-in.-sec2
- k :
-
torsional stiffness=πd 4 E s /32l, lb-in./rad
- l :
-
length of drive shaft, in.
- t :
-
time, sec
- W :
-
weight of rotor, lb
- β:
-
a dimensionless parameter =\(16 Wl/\pi d^3 E_s \times \left( {\mu _s - \mu _k } \right)\), dimensionless
- δ:
-
logarithmic decrement, dimensionless
- ζ:
-
damping ratio =\(c/\left( {2\sqrt {kI} } \right)\), dimensionless
- θ:
-
angular position of rotor, rad
- θ i :
-
angular position of input end of shaft, rad
- μ k :
-
coefficient of kinetic friction, dimensionless
- μ s :
-
coefficient of static friction, dimensionless
- ψ:
-
\(\theta - \theta _i \), angular position of rotor relative to input end of shaft, rad
- ψ:
-
\(\dot \theta - \omega \), angular velocity of rotor relative to constant velocity at input end of shaft, rad/sec
- \(\ddot \psi \) :
-
\(\ddot \theta \), angular acceleration of rotor, rad/sec2
- ω:
-
constant input velocity, rad/sec
- ω c :
-
critical input velocity which produces instability, rad/sec
- \(\omega _n \) :
-
natural frequency (undamped) =\(\sqrt {k/I} \), rad/sec
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Kemper, J.D. A study of system parameters affecting self-excited torsional oscillations. Experimental Mechanics 6, 342–349 (1966). https://doi.org/10.1007/BF02327210
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DOI: https://doi.org/10.1007/BF02327210