referring now to fig1 , the shaft 1 supporting the double - inlet centrifugal impeller 2 has a tapering profile from a widest part at the center of the rotor to a narrowest part near each end of the shaft . this is a typical shaft design for large double inlet fans , but parallel solid shafts or hollow shafts can also be used . the shaft 1 is driven by an electric motor 5 shown only schematically in fig1 . the air or other gas being pumped by the fan enters the interior of the fan from the left - hand side and from the right - hand side of the double - inlet impeller and exits radially outwardly , with a circumferential flow component , so the difference between the external periphery of the shaft 1 at its wider diameter central region and the internal diameter of the impeller 2 surrounding that portion of the shaft defines the cross - sectional area for gas being pumped by the fan impeller and must be as large as is possible . since the internal diameter of the impeller is fixed by its geometric proportions this can only be achieved by minimizing the diameter of the shaft . with the conventional fan operating system the rotor shaft bearings 4 a and 4 b shown schematically in fig1 must be strong enough to hold the shaft ends so as to support the combined weight of the shaft 1 and the fan impeller 2 , but also so as to be able to absorb any tendency for lateral vibration of the impeller shaft , for example a tendency to revolve about the geometric axis 6 of the shaft when the impeller is being run at its normal operating speed , or within its normal operating speed range , so that the first critical speed is kept as high as possible and therefore well above the normal operating speed , or normal operating speed range . fig2 shows the vibration amplitude increasing to a maximum amplitude of one unit at the rotational speed of 600 revolutions per minute in the case where the critical speed of the fan and shaft assembly is 750 revolutions per minute , i . e . well above the fan maximum running speed . if this design margin shown in fig2 were not available , the fan impeller and shaft assembly would undergo dangerous transverse vibration even at the maximum running speed of the fan . the arrangement shown in fig1 and 2 is that of a conventional double - inlet centrifugal fan . fig3 shows the system modified in accordance with the present invention where the shaft no longer needs to be so stiff and can more readily be of non - tapering form , thereby allowing a smaller diameter central region of the shaft 11 and thereby increasing the inlet area for a given impeller internal diameter . nevertheless the strength of the bearings 14 a and 14 b of the rotor shaft can be the same as for fig1 . in the case of the fan of fig3 there is a surprising difference in that the fan needs to be run up through the first critical speed in order to attain its normal operating speed or its normal operating speed range , and the design margin , which in fig1 represents the difference in rotational speeds between the relatively high first critical speed and the lower speed normal operating range , is reversed in that in the case of fig4 that design margin is the difference between the relatively low first critical speed and the minimum running speed which is now higher than the first critical speed . the fan motor 15 will have a power rating adequate to provide the maximum power absorbed by the fan at its maximum operating speed . it is envisaged that the fan rotor will thus be accelerated rapidly through its first critical speed so that no problems of transverse vibration will be encountered during run up . however , if the fan is allowed to coast down through its first critical speed it is likely to have a deceleration value the modulus of which is less than the acceleration of the fan on run up , and in such a situation it may be desirable to provide an optional brake for decelerating the fan impeller more rapidly to reduce the duration of exposure to transverse vibrations as it passes down through the first critical speed band . however it is not expected that the use of a brake will in itself reduce the amplitude of those transverse vibrations . the required braking can be provided by various means , such as a shaft mounted disc brake or electrical braking acting directly on the main drive motor . fig5 a shows the run - up condition when the fan running speed is being increased through the critical speed to the maximum speed , with the speed plotted against time so as to illustrate the very short time interval during which the impeller is accelerating through the first critical speed . this short time interval results from the asymptotic curve in fig5 a . fig5 b shows a plot of vibration amplitude ( measured by units 1 to 12 ) against the rotational speed and shows that the rate of increase of vibration is slightly increased from about 75 % of first critical speed up to that peak and that the curve is a mirror image during drop - off of vibration amplitude during increase beyond the first critical speed . the normal fan running speed is in this case approximately 30 % above the first critical speed , and at this speed an adequate safety margin is provided . fig6 a shows another plot of running speed against time , corresponding to fig5 a , and shows that in the absence of braking there can be a considerable time interval during which the rotational speed is at or near its first critical speed value during free running - down of the fan . fig6 b , plotting the vibration amplitude against rotational speed shows the considerable increase in fan vibration , to a peak of more than 12 units , during the ( coasting ) run - down in the absence of braking . as indicated above , by having an optional rotor brake to reduce the fan impeller speed much more rapidly during run - down there will be a much shorter exposure to the vibrations resulting from passing through the first critical speed band , and the result of this will be to lower the peak of the curve in fig6 b , in order to bring the degree of vibration amplitude experienced during run - down down to somewhat nearer the maximum vibration amplitude experienced during run - up ( fig5 b ). as indicated above , the fan operating system modified in accordance with the present invention preferably includes a motor capable of rapid acceleration through the first critical speed value and may include a rotor brake capable of reducing the time interval when the rotational speed of the rotor is close to its first critical value . the present invention provides not only a new design concept in enhancing the operation of a fan , but also a way of more cheaply improving the aerodynamic performance of a fan installation by allowing the use of smaller shaft bearings with a narrower diameter shaft with a larger diameter fan impeller so as to provide for a much greater diameter difference between the internal diameter of the impeller and the external diameter of the shaft , and hence to provide for a much greater cross - sectional area of pumped gas flow than for a conventional fan . the term “ flexible shaft ” used within the present application generally implies that the first critical speed of the assembly of fan impeller and shaft is lower than the normal operating speed of that fan . it will of course be understood , in the context of retrofitting a fan by the present invention ( fig3 and 4 ) in place of the conventional fan of fig1 and 2 , it is possible to replace the original impeller by a smaller impeller running at a higher rotational speed ( above the first critical speed ) and nevertheless still to achieve an enhanced aerodynamic performance of the retrofit fan . this would not be possible if the critical speed must always be above the maximum running speed , and therefore represents a significant advantage offered by the present invention . such a smaller impeller can also be cheaper to produce and , taking into account the ability to operate on the original shaft bearings , no longer requiring strengthening , this provides a considerable economic advantage . fig7 a , 7 b and 7 c show an example of a retrofit situation in the case of a double - inlet centrifugal fan in which the impeller / shaft combination in accordance with the present invention is shown at fig7 c as a modification of the original installation shown in fig7 a , and fig7 b shows what would have been carried out if traditional thinking had been followed . the installation of fig7 b is therefore not in accordance with the present invention . the arrangement shown in fig7 a has an impeller 22 a operating at a design speed of 890 rpm with a stiff rotor shaft 21 a supported at each end by bearings 24 a and 24 b , respectively . these bearings are themselves supported on upstanding arms of a foundation assembly 23 which also supports a fan drive motor 25 . fig7 a also shows , as plots of fan static pressure in kilopascals ( kpa ) against volumetric flow rate in m 3 / s , ( i ) the full output fan characteristic , ( ii ) the design system resistance curve which intersects the full output fan characteristic at the maximum operating duty of the fan of fig7 a , and also ( iii ) the desired new operating duty of the fan , resulting in an increase in the rotational speed to 1185 rpm . fig7 b shows this higher operating speed fan assembly with the conventional stiff rotor shaft 21 b supporting the impeller 22 b which can be smaller than impeller 22 a of fig7 a , by virtue of the increased rotational speed . however , fig7 b also shows that the shaft 21 b of this modification needs to be larger in diameter to ensure that the first critical speed is above the higher running speed of the replacement fan . consequently the diameter of the shaft 21 b in the vicinity of the impeller 22 b is considerably larger than was the case for the original fan of fig7 a , and the fluid inflow path suffers considerable chokage which will prove unacceptable to the fan operation . on the other hand , fig7 c again shows an arrangement in which the impeller is rotating at 1185 rpm , and the maximum diameter of the impeller 22 c of fig7 c is the same as that of impeller 22 b of fig7 b ( but smaller than that of fig7 a ), but the impeller 22 c differs in that it is supported on a flexible , smaller diameter shaft 21 c which allows the area of inlet to the impeller to be substantially greater and hence provides an improved full output fan characteristic which is also shown in fig7 c . the new operating duty of the fan , again defined by the intersection of the design system resistance curve and the full output fan characteristic , is now much higher at around 8 kpa as compared with 5 kpa for the fan of fig7 a . as a result , the modification required to replace the impeller 22 a and shaft 21 a of fig7 a by the smaller diameter impeller 22 c and flexible shaft 21 c of fig7 c can be achieved during a normal plant shutdown period while other essential maintenance is performed , thereby avoiding lost commercial operating time . although fig1 to 7 are related to a double - inlet centrifugal fan , the present invention can also be applied to a single - inlet centrifugal fan or to a single - inlet fan employing axial flow , or to single - inlet fan employing mixed centrifugal and axial flow . fig8 a shows such a single - inlet fan employing mixed axial and centrifugal flow where the fluid approaching the rotor 32 a follows an axial flow path and undergoes a partial alteration into radially inward and axial flow , following which it is deflected radially outwardly by deflector blades 36 to be discharged along a direction perpendicular to the access of the fan rotor 31 . fig8 b shows a generally similar fan arrangement , but where the fan rotor 32 b merely induces axial flow in the incident air , then discharging it axially to be deflected along a path perpendicular to the axis of rotation of the rotor shaft 31 . although not shown in the drawings , the present invention can also be used with a single inlet centrifugal fan where effectively the rotor is one half of the fan impeller shown in fig1 and 3 . the advantages of the present invention with relation enhancement of a double - inlet centrifugal fan have been well described and illustrated above as offering both the need for unreinforced bearing mountings and also the opportunity of providing a larger cross - sectional flow passage of the air . in the case of the single - inlet fan the effect of the shaft redesign on the aerodynamic flow path is much less or even negligible , as can be seen from fig8 a and 8 b where the annular airflow path is spaced radially outwardly from the cylindrical surface of the shaft 31 . nevertheless , the advantages of being able to avoid strengthening of the bearing foundations when upgrading to a higher fan rotor speed can be appreciated both with the single - inlet centrifugal fan type and with the mixed flow and axial flow types such as are illustrated in fig8 a and 8 b .