Shafts critical speed

The calculations in sections 13.3 and 13.5 are based on avoidance of the shaft critical speed or sudden failure due to jamming. This section allows the basic designs in those sections to be assessed for the probability of failure due to fatigue. This is expressed in terms of a factor of safety for fatigue, Ff. 

Expander-compressor shafts are preferably designed to operate below the first lateral critical speed and torsional resonance. A flame-plated band of aluminum alloy or similarly suitable material is generally applied to the shaft in the area sensed by the vibration probes to preclude erroneous electrical runout readings. This technique has been used on hundreds of expanders, steam turbines, and other turbomachines with complete success. Unless integral with the shaft, expander wheels (disks) are often attached to the shaft on a special tapered profile, with dowel-type keys and keyways. The latter design attempts to avoid the stress concentrations occasionally associated with splines and conventional keyways. It also reduces the cost of manufacture. When used, wheels are sometimes secured to the tapered ends of the shaft by a common center stretch rod which is pre-stressed during assembly. This results in a constant preload on each wheel to ensure proper contact between wheels and shaft at the anticipated extremes of temperature and speed. 

The standards define terms used in the industry and describe the basic design of the unit. It deals with the casing, rotors and shafts, wheels and blades, combustors, seals, bearings, critical speeds, pipe connections and auxiliary piping, mounting plates, weather-proofing, and acoustical treatment. 

Critical speeds of a turbine operating below its first critical should be at least 20% above the operating speed range. The term commonly used for units operating below their first critical is that the unit has a stiff shaft, while units operating above their first critical are said to have a flexible shaft. There are many exciting frequencies that need to be considered in a turbine. Some of the sources that provide excitation in a turbine system are ... 

The previous equation shows that when lu < uj ,8r is positive. Thus, when operating below the critical speed, the system rotates with the center of mass on the outside of the geometric center. Operating above the critical speed (lu > LUn), the shaft deflection 8r tends to infinity. Actually, this vibration is damped by outside forces. For very high speeds (lu >> LUn), the amplitude 8r equals —e, meaning that the disc rotates about its center of gravity. 

The critical-speed calculation of a rotating shaft proceeds with equations to relate loads and deflections from station — 1 to station n. The shaft shear V can be computed using the following relationship ... 

This speed becomes critical when the frequency of excitation is equal to one of the natural frequencies of the system. In forced vibration, the system is a function of the frequencies. These frequencies can also be multiples of rotor speed excited by frequencies other than the speed frequency such as blade passing frequencies, gear mesh frequencies, and other component frequencies. Figure 5-20 shows that for forced vibration, the critical frequency remains constant at any shaft speed. The critical speeds occur at one-half, one, and two times the rotor speed. The effect of damping in forced vibration reduces the amplitude, but it does not affect the frequency at which this phenomenon occurs. 

The undamped critical speed is proportional to the static deflection nf a sii shaft as seen by the following equation for a mass concent at a le point [6],... 

Nf. = critical speed, rpm < = siatic shaft deflection L, = L ravitational constant... 

Critical speed the mixer shaft speed which matches the first lateral natural frequency of the shaft and impeller system. Excessive vibrations and shaft deflections are present at this speed. 

The mixer manufacturer should always be consulted for proper mechanical features design and strength characteristics, such as horsepower, gear rating AGA, shaft diameter, shaft deflection, critical speeds, bottom steady bearing, and side shaft bearings. 

The shaft is a forging and may he designed as a stiff shaft or flexible shaft. A stiff shaft design means that the shaft will operate helow any of its critical speeds. Usual practice limits design operation to 60% of the first critical speed. This requires a heavier shaft than the flexible design that allows the shaft to pass through its first critical speed at 40-60% of normal and maximum operating speeds. 

All turbines are variable-speed drivers and operate near or above one of the rotor s critical speeds. Narrowbands should be established that track each of the critical speeds defined for the turbine s rotor. In most applications, steam turbines operate above the first critical speed and in some cases above the second. A movable narrowband window should be established to track the fundamental (1 x), second (2x), and third (3x) harmonics of actual shaft speed. The best method is to use orders analysis and a tachometer to adjust the window location. 

Bushings shall be suitably corrosion-resistant and abrasion-resistant for the specified product and temperature. The maximum spacing between shaft bushings shall be in accordance with Figure 32 in order to maintain the first critical speed above the maximum allowable continuous speed. 

A typical formula for calculating the first natural frequency (critical speed) of an agitator shaft considers the shaft stiffness, the shaft length, the weights of impellers and shaft, and the rigidity of the shaft mounting ... 

The stated operating speed of 100 r/min is only 59 percent of this critical speed, so the 3.0-in shaft should operate safely. 

The critical-speed problem may in some cases be solved hand in hand with a common problem related to dynamic loads. One source of dynamic loads on an agitator shaft is the waves and vortices that occur when an impeller operates near the liquid surface, such as when a tank fills or empties. 

Another alternative to avoid critical-speed problems is the use of a shorter shaft. Reducing the shaft length and impeller extensions by 10 in (0.25 m) reduces equivalent weight to... 

A complete mechanical design should include a calculation of the critical speed. When the operating speed is close to or exceeds the critical shaft speed, vibrations can lead to mechanical instabilities and severe equipment damage. For more information, see the chapter by King [6] in Mixing in the Process Industries or the chapter by Dickey and Fasano in the Handbook of Industrial Mixing [1]. 

The final check is to ensure that the shaft is not running at or near its critical speed. This is explained in the following sections. 

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