Reciprocating Systems Vibration

Torsional oscillations, in the crankshaft and in the shafts of driven machinery, is the vibration phenomenon of practical importance in the design of reciprocating engines.

The average torque delivered by a cylinder in a reciprocating machine, is a small fraction of the maximum torque, which occurs during the firing period.

Even though the torque is periodic the fact that it fluctuates so violently within the period, constitutes one of the inherent disadvantages of a reciprocating machine, from the dynamics point of view, as compared with a turbine where the torque is practically uniform.

It is possible to express the torque by a reciprocating engine into its harmonic components of several orders of the engine speed, and these harmonic components can excite the engine driven installations into forced torsional vibrations.

The engine and the driven unit such as generator or a pump are normally connected by a flexible coupling and thus the total installation has fairly low natural frequencies falling in the speed range of the engine and harmonics of different order.

It is a commonly known fact that failures can occur in reciprocating machine installations, when the running speed of the engine is at or near a dominant torsional critical speed of the system.

High dynamic stresses can occur in the main shafting of such engine installations and to avoid these conditions, it is essential that the torsional vibration characteristics of the entire installation be analysed before the unit is put into operation.

Any analysis of torsional vibration characteristics of the reciprocating machinery should finally predict the maximum dynamic stresses or torque developed in the shafting & couplings of the system, as accurately as possible, so that they can be compared with the permissible values, to check the safety of installation.

Modelling of reciprocating machine systems:

A multi-cylinder reciprocating machine contains many reciprocating and rotating parts such as pistons, connecting rods, crank shafts, flywheels, dampers and cranks. The system is so complicated that it is difficult, if not impossible, to undertake an exact analysis of its vibrational characteristics. The actual system is characterised by the presence of unpredictable effects like variable inertias, internal dampings, misalignments in the transmission units, uneven firing order etc. The analysis can be best carried out, by lumping the inertias of the rotating and reciprocating parts at discrete points on the main shaft. The problem then reduces to the forced torsional vibration study of an N disc system subjected to varying torques at different cylinder points. The crankshaft and the other drive or driven shafts are generally flexible in torsion, but have low polar moments of inertia, unlike in the case of some large turbines or compressors. On the other hand, the parts mounted on the shafting, like the damped,flywheel, generator etc. are rigid and will have very high polar mass moments of inertia. The system containing the crankshaft, coupling, generator/auxiliary drive shaft/ other driven shaft like pumps, and mounted parts can then be reduced to a simple system with a series of rigid discs (representing the inertias) connected by the mass less flexible shaft as shown in Figure 1.

(i) Polar mass moments of inertia

Determination of polar mass moments of inertia is a straightforward matter for rotating parts; however, it is not simple in the case of reciprocating parts. Consider the piston shown in two different positions in Figure 2, and imagine the crankshaft with a polar mass moment of inertia,   to be non-rotating but executing small torsional oscillations. In the first case (Figure 2(a)) there is no motion for the piston, with small oscillations of the crank and hence the equivalent inertia of the piston is zero. Whereas in the second case (Figure 2(b)), the piston has practically the same acceleration as that of the crank pin and the equivalent inertia is , where is the mass of reciprocating parts and r is the crank radius.

Hence, the total polar mass moments of inertia varies from   to , when the crankshaft is rotating.

The inertia of the connecting rod can be taken care of by considering dynamic equivalent two masses: one at the piston & other at the crank pin. It will contribute to both and by a small amount.

We consider, as an approximation, the system to have an average inertia given by



Where r is crank radius.

 
Torsional stiffness of shafts connecting discs

In determining the torsional stiffness of the shafts connecting rotors, the main difficulty arises from the crank webs.

 
Considering an idealisation of a crank shaft to an ordinary straight shaft having the same flexibility as shown

In Figure 3. Through this idealisation is physically possible, but the calculations involved are extremely difficult. This is because the crank webs are subjected to bending and the crank pin to twisting, when the main shaft is subjected to twisting. Moreover the beam formulae, if used will not be very accurate, because of short stubs involved rather than long beams usually considered. Further torques applied at the free end also gives rise to sidewise displacement, which is prevented in the machine. For high-speed lightweight engines, the crank webs are no more rectangular blocks and application of the theory becomes extremely difficult. Because of this uncertainties in analytical calculation to estimate the torsional stiffness of crank shafts, several experiments have been carried out on a number of crank shafts of large slow speed engines, which have shown that the equivalent length le is nearly equal to the actual length, if the diameter of main shaft is equal to the crank pin diameter. In general the procedure that is applied to reduce the reciprocating machine system to a mathematical model, is to use a basic diameter, which corresponding to the journal diameter of the crankshaft. The torsional stiffness are all calculated based on the basic diameter, irrespective of their actual diameter. For the end rotors (i.e. the generator rotor) compute the stiffness of the shaft from the coupling up to the point of rigidity (since generator rotors are relatively rigid so shaft from coupling to main generator rotor need to be considered). In case where one part of the system is connected to the other part through gears, or other transmission units, it is convenient to reduce all the inertias and stiffness to one reference speed. Once the mathematical model is developed, it can be used for illustrating the critical speed calculations and forced vibration response. Use of FEM can give more accurate stiffness of such irregular shaped shafts.

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