The shaft of all types of fan may be treated as a beam carrying the impellers as point loads if the shaft is long, or as a thickening of the shaft if it is short. The bearings, especially if self-aligning, are treated as simple supports. Only in the old-fashioned sleeve bearings, where the journal might be 3 diameters long, was it possible to consider them as approaching rigid encastrй supports.
The shaft must be considered for three different criteria and that giving the largest diameter must be taken as the basis of the design:
• Maximum sheer stress
• Maximum direct stress
• Critical speed
In order to carry out these calculations, it will be necessary to fix the type, size and position of the bearings (see Chapter 10). Where the fan is driven through vee belts (see Chapter 11 ) the belt tension will give an additional load which is used for calculating stresses. It should not however be used for critical speed determination as, unlike out-of-balance, it is unidirectional.
Bending stresses result from the overhang effects of the impeller and from the moment produced by the belt pull in indirect drive units. Torsion results from the work done by the fan in rotating at the speed necessary to achieve the duty. If the system resistance is lower or higher than that specified, this will affect the power absorbed and thus the torque required. It may also affect the belt pull in indirect drive units and thus the bending stress.
Max direct stress f is:
F = M+Vm2 + T2^) Equ7.9
Max shear stress q is:
Q = -^r a/m2 + T2 Equ 7.10
M = maximum bending moment
T = maximum torque
Ds = shaft diameter
All in consistent SI units.
The acceptable stresses will be determined by the shaft material, whilst the maximum bending moment and torque are determined by the arrangement of impeller, bearing centres and belt pull, etc.
It is essential to allow reasonable factors of safety on the maximum stresses attained to cater for the effects of unbalance, additional accelerating torque at start-up, fatigue, over tightened vee belts etc.
As the rotational speed of a fan is increased, it will be seen that at certain speeds the shaft may vibrate quite violently whereas at speeds above and below these it will run relatively quietly. The speeds at which these severe vibrations occur are known as the critical speeds of the rotating assembly.
If a unit operates at or near a critical speed, large amplitudes of vibration can be built up. Such a condition results in dangerously high stresses, possible rubbing of the impeller eye on the inlet cone, and large cyclical forces transmitted to the foundations. It is therefore important that there is a margin between the running and critical speed.
Many textbooks suggest that this margin should be a minimum of 20%. The author suggests however that for all non-symmet — rical arrangements, i. e. all single inlet fans, the ratio of critical speed should be at least 1.5. This ratio is a measure of the shaft stiffness and determines the dynamic effect of unbalance. For a given system it can be shown that the eccentricity of the centre of gravity of the impeller is increased by 80% for a ratio of 1.5 but only 20% when the ratio is 2.5. The disturbing forces, which have to be resisted by the bearings, bearing supports and ultimately the foundations, increase in proportion to the eccentricities.
Where fans are handling large quantities of foreign matter and are thus subject to build-up, erosion, corrosion or temperature
Distortion, a minimum ratio of— of 1.8 is recommended.
For double inlet fans, due to the symmetricity, the ratio for clean air fans may be reduced to 1.3.
Ratios close to 2 should however be avoided as they may coincide with the second harmonic of critical speed.
It can be shown that all critical speeds are:
Misalignments, the passing of the impeller blades by the casing cut-off or tongue piece, or by rapid fluctuations in system resistance. If the frequency of these impulses coincides with, or is a multiple of, the torsional critical speed, then large amplitude oscillations may build up and a possible shear fatigue failure occur.
Most fan installations will have onlytwo masses the fan impeller and the motor rotor for which the frequency F:
F-JL I1 PE*(J1 + J2)
2 71 ’
Natural frequency (Hz)
Polar moment of inertia of shaft (m4)
Shaft diameter (m) shear modulus of elasticity (Pa) shaft length between masses (m) mass moment of inertia = mr2 mass of impeller or rotor (kg) radius of gyration (m)
The formula becomes very much more complex for a stepped shaft.
Ds = shaft diameter (m)
M = impeller mass (kg)
I = distance from impeller e. g.
To supporting bearing (m)
The actual values will depend on the fan arrangement, bearing centres, overhang of impeller etc. This formula is therefore a simplification but does show which factors are of importance.
It should be noted that perfect balance of an impeller and shaft is impossible. There is always a residual unbalance however small. Rotation produces a centrifugal force of the mass centre which is balanced by the springing action of the shaft.
Below the first critical speed, the centre of gravity (c. g.)of the impeller and shaft assembly rotates in a circle about the geometrical centre, whereas above the first critical speed the shaft rotates about the e. g. This leads to extremely smooth running and is the “norm” for turbo-generators. There are now engineers advocating its use for large fans especially where the impeller is between bearings and the blockage effects of the shaft are severe. It does of course require that the fan rapidly accelerates through the critical speed.
The axis of rotation changes at the critical speed from the geometric centre to the centre of gravity. When the shaft rotates at critical speed the restoring force of the shaft is neutralised and the action is dynamically unstable, hence large amplitudes of vibration may occur.
In addition to the lateral critical speeds described in Section
7.9.3 there are torsional critical speeds where two or more rotating masses are connected by a shaft. These must be avoided for trouble-free running.
As a fan impeller rotates, small torque impulses may develop and be transmitted to the shaft. They may be caused by slight
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