Critical speed calculations

Critical Speed Calculations for Rotor Bearing Systems... 

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 ... 

Critical Speed. The first lateral critical speed of the rotating assembly shall be at least 120% of die maximum operating speed. A dry critical speed calculation (see HI 9.6.4) is adequate to verify compliance. HI 9.6.4 shall be used to calculate static deflections used for the critical speed calculation. 

If the shaft speed is near the natural frequency, increase the solid shaft size to the next in. larger, or metric equivalent. Then redo the critical speed calculation. Again, if the operating/critical speed ratio meets the natural... 

The minimum shaft diameter will be the greater of the two values calculated in eqs. (21-5) and (21-6). For practical purposes, most mixer shafts are made from bar stock, so standard sizes are nsually available in or 1 in. increments or certain multiples of millimeters. For critical speed calculations, the next larger standard shaft diameter shonld be nsed. 

A hollow shaft, made from pipe, can increase the stiffness and reduce the weight (mass) of a mixer shaft in critical speed calculations. Such changes will increase the natnral freqnency and extend the allowable shaft length or operating speed. When determining the appropriate shaft size for the strength of a hollow shaft, begin with the dimensions for standard available pipe or tnbe. Then compute the shear and tensile stress values and compare them with the allowable values. The equations for combined shear and tensile limits in hollow shafts are, respectively. 

Methods for calculating undamped and damped critical speeds that closely follow the works of Prohl and Lund, respectively, are listed herein. Computer programs can be developed that use the equations shown in this section to provide estimations of the critical speeds of a given rotor for a range of bearing stiffness and damping parameters. 

Prohl, M.A., General Method of Calculating Critical Speeds of Flexible Rotors, Trans. ASME, J. Appl. Mech., Vol. 12, No. 3, September 1945, pp. A142-A148. 

The torsional resonant response of a system is an interaction of all the components in the train. Calculation of torsional natural frequencies is based on the entire system and these frequencies are valid only for that given arrangement. If any component of the train is replaced by an item with torsional characteristics different from the original, the system tor sional response must be recalculated and new torsional natural frequencies determined. Occasionally, an original equipment manufacturer is requested to calculate the torsional and lateral critical speeds of the supplied item. Unfortunately, the purchaser is unaware that this request is of limited value since the torsional response of a single item in a train is meaningless. Likewise, a torsional shop test will yield meaningless results if the train is not assembled and tested with every item destined for the field. 

Rotor-bearing system dynamic stability and calculated critical speed... 

It can be seen from equation 13.136 that the critical speed of a centrifuge will depend on the mass of the bowl and the magnitude of the restoring force it will also depend on the dimensions of the machine and the length of the spindle. The critical speed of a simple system can be calculated, but for a complex system, such as loaded centrifuges, the critical speed must be determined by experiment. It can be shown that the critical speed of a rotating system corresponds with the natural frequency of vibration of the system. 

Note The principal objective of shop verification by response to unbalance is to verify the existence of a critical speed (vibration peak) within tolerance of the calculated value, or if the analysis predicted a highly damped critical speed, the absence of a vibration peak within tolerance of calculated value. Shop verification by this method is feasible only for pumps that have sleeve bearings and are furnished with proximity probe pairs at each journal bearing. 

Note Highly damped critical speeds will not be observable therefore, the absence of rotor response in the region of a calculated highly damped critical speed will be verification of the analysis. 

Calculate the critical speed of a mill 28 in. in diameter. What is the significance of critical speed in grinding operations ... 

In a gas-inducing reactor, both gas and liquid phases are generally considered to be completely backmixed. The use of Eqs. (2.39) and (2.40) for the calculations of the critical speed for gas induction is recommended. The rate of gas induction can be expressed by a dimensionless relation NA = /(FrdJH, Ga, dT/du HJdj). The most important parameters are Fr dJH and dT/di. For a given power input per unit volume, the turbo aerator appears... 

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 ... 

Calculate the percent of critical speed at which the mill should be operated. This follows directly from steps 3 and 4. Thus, percent of critical speed equals 100(13.8)/22.1 = 62.4 percent. 

Mill rotation speeds are set such that the balls are lifted nearly to the top of the jar before tumbling to the bottom. As a rule, about 50-75% of the critical speed or the speed at which the media are held to the walls by centrifugal force is appropriate. The critical speed (in rpm) can be calculated as TTij r, where r = jar radius (cm). 

For actual applications the drum speed is normally kept between 25 and 40% of the critical speed which can be calculated from... 

As with balling drums, it is most important to select the correct rotational speed of the granulation pan. It is defined as the percentage of critical speed that can be calculated by (see Figure 112)... 

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]. 

---