02-01-2013, 11:09 AM
TORSIONAL VIBRATIONS OF ROTOR SYSTEMS
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INTRODUCTION
Torsional vibrations is predominant whenever there is large discs on
relatively thin shafts (e.g. flywheel of a punch press).
Torsional vibrations may originate from the following forcing
Inertia forces of reciprocating mechanisms (such as pistons in IC
engines)
Impulsive loads occurring during a normal machine cycle (e.g. during
operations of a punch press)
Shock loads applied to electrical machinery (such as a generator line
fault followed by fault removal and automatic closure)
Torques related to gear mesh frequencies, turbine blade passing
frequencies, etc.
For machines having massive rotors and flexible shafts (where the
system natural frequencies of torsional vibrations may be close to, or
within, the excitation torque frequency range during normal operation)
torsional vibrations constitute a potential design problem area.
In such cases designers should ensure the accurate prediction of
machine torsional frequencies and frequencies of any torsional load
fluctuations should not coincide with the torsional natural frequencies.
Hence, the determination of torsional natural frequencies of the system is
very important.
Simple System with Single Rotor Mass
Consider a rotor system as shown Fig.1
The shaft is considered as massless and it provides torsional stiffness only.
The disc is considered as rigid and has no flexibility.
If an initial disturbance is given to the disc in the torsional mode and
allow it to oscillate its own, it will execute the free vibrations as shown in
Fig. 2.
It shows that rotor is spinning with a nominal speed of w and executing
torsional vibrations, q(t), due to this it has actual speed of {w + q(t)}.
It should be noted that the spinning speed remains same however angular
velocity due to torsion have varying direction over a period.
The oscillation will be simple harmonic motion with a unique frequency,
which is called the torsional natural frequency of the rotor system.
MODF Systems
When there are several number of discs in the rotor system it becomes is
multi-DOF system. When the mass of the shaft itself may be significant
then the analysis described in previous sections (i.e. single or two-discs
rotor systems) is inadequate to model such system, however, they could
be extended to allow for more number of lumped masses (i.e. rigid discs)
but resulting mathematics becomes cumbersome.
Alternative methods are:
o Transfer matrix methods
o Methods of mechanical impedance
o Finite element methods
Transfer matrix method:
• A multi-disc rotor system, supported on frictionless supports, is shown in
Fig. 7. Fig. 8 shows the free diagram of a shaft and a disc, separately. At
particular station in the system, we have two state variables: the angular
twist q and Torque T.
Geared Systems
In some machine the shaft may not be continuous from one end of the
machine to the other, but may have a gearbox installed at one or more
locations. So shafts will be having different angular velocities as
shown in Fig.15(a).
For the purpose of analysis the gear system must be reduced to
system with a continuous shaft so that they may be treated as
described in the preceding section.