Natural frequency, shaft design

Near top speed, a fan may operate at a speed that is near or above the natural frequency of the wheel and shaft. Under such conditions, the fan can vibrate badly even when the wheel is clean and properly balanced. Whereas manufacturers often do not check the natural frequency of the wheel and shaft ia standard designs, many have suitable computer programs for such calculations. Frequency calculations should be made on large high speed fans. The first critical wheel and shaft speed of a fan that is subject to wheel deposits or out-of-balance wear should be about 25—50% above the normal operating speed. 

Shaft design must accommodate hydraulic and mechanical loads and must avoid vibration near the natural frequency. A typical overhung shaft arrangement with dimensional nomenclature is shown in Figure 21-32. Hydraulic loads on the shaft result from the torque required to turn the impeller(s) and random or systematic lateral hydraulic loads on the impeUer(s). Other sections of the book describe methods for determining impeller power. Shaft design will use impeller power to calculate torque and hydraulic forces and thus size a shaft within allowable stress limits. 

The steps necessary to design a mixer shaft first consider strength, then commercially available material, and finally, natural frequency. The following steps also consider alternatives if natural frequency problems are encountered. 

For this standard shaft size, determine the natural frequency of the shaft and impeller system. If the system meets the natural frequency criterion, the design is complete. 

In the following sections we present methods for calculating strength and natural frequency for different shaft types. All these methods make certain assumptions about the design. Other methods using other assumptions can be developed and used but usually give similar results. 

Natural frequency is a dynamic characteristic of a mechanical system. Of primary concern to mixer design is the first lateral natural frequency, which is the lowest frequency at which a shaft will vibrate as a function of length and mass. The first lateral natural frequency is analogous to the vibration of a tuning fork, except on a larger scale. 

Transmissibility is also often called the force magnification factor. Any force applied to a shaft under dynamic conditions will be amplified by this magnification factor. A side load of 100 units at rest for a damping ratio of 0.1, will behave as a 257 unit side load when N/Nc = 0.8 and as a 388 unit side load when N/Nc = 0.9. Most mixer manufacturers use design stress limits based on an allowable approach to the first natural frequency, Nc. The worst-case scenario... 

The other key design assumption is that the support stiffness is sufficiently large that the overall stiffness, k, is controlled only by the shaft stiffness. With a stiff support the natural frequency depends only on the shaft stiffness and associated mass. Mixer manufacturers generally assume that the mixer will be mounted on a strncture where a small change in stiffness does not significantly affect the natnral frequency. 

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