Dynamical balance

The reactivities of tlie species witliin tlie Wilkinson cycle are so great tliat tliey are not observed directly during tlie catalytic reaction ratlier, tliey are present in a delicate dynamic balance during tlie catalysis in concentrations too low to observe easily, and only tlie more stable species outside tlie cycle (outside tlie dashed line in figure C2.7.2 are tlie ones observed. Obviously it was no simple matter to elucidate tliis cycle tlie research required piecing it togetlier from observations of kinetics and equilibria under conditions chosen so tliat sometimes tlie cycle proceeded slowly or not at all. 

A bent shaft is physically bent and distorted. Placing the shaft into a lathe or dynamic balancer and rotating it will reveal the distortion. If a bent shaft is installed into a pump and run, it will fail prematurely, leaving evidence and specific signs on the eircumferenee of close tolerance stationary parts around the pump s volute circle. The shaft will exhibit a wear spot on its surface where the elo.se tolerance parts were rubbing. 

A deflected shaft is absolutely straight when rotated in a lathe or dynamic balancer. The deflection is the result of a problem induced either by operation or. system design. The deflected shaft also will fail prematurely in the pump, leaving similar, but different evidence on the elo.se tolerance rubbing parts in the pump. The next two pictures show how a bent shaft appears when rotated 180 degrees (Figure 9-9 and Figure 9-10). 

Next we ll discuss evidence marks and prints that are different, but to the untrained eye, they may appear the same. You may see a spot or arc of wear and gouging on the rotary elements, and a eireumferential wear circle on the bore of the close tolerance stationary elements. This is a maintenanee-indueed problem, d his is the sign of a physically bent shaft, or a shaft that is not round, or a dynamic imbalance in the shaff-sleeve-impeller assembly. The solution is to put the shaft on a lathe or dynamic balancer, verify its condition, and correct before the next installation. 

The problem could be maintenance, operation, or design, or a combination of any or all these factors. In all honesty, you should never. see this set of evidence marks because it indicates a lack of control. Now because the mechanic cannot control operational problems or design problems, the first phase to correct this situation is to control the mechanical maintenance factors, like alignment, proper bolting and torque. sequences, be sure shafts are straight and round, and dynamically balance all rotary components. Reinstall the pump and wait for the next failure. Once the maintenance factors are under control, there should appear a clear vision and path to resolve any operational and/or design weaknesses. 

Another concept that has become quite popular in industry is the balanced mechanical seal. Most manufacturers offer seal models incorporating the balance feature. This balance is not a dynamic balance, but instead a relationship between the forces tending to open the faces in a mechanical seal and the forces tending to close the seal faces (Figure 13-8). 

Gear elements must be multiplane and dynamically balanced. Where keys are used in couplings, half keys must be in place. The maximum allowable unbalanced force at maximum continuous speed should not exceed 10% of static weight load on the journal. The maximum allowable residual unbalance in the plane of each journal is calculated using the following relationship... 

Flexible rotors are designed to operate at speeds above those corresponding to their first natural frequencies of transverse vibrations. The phase relation of the maximum amplitude of vibration experiences a significant shift as the rotor operates above a different critical speed. Hence, the unbalance in a flexible rotor cannot simply be considered in terms of a force and moment when the response of the vibration system is in-line (or in-phase) with the generating force (the unbalance). Consequently, the two-plane dynamic balancing usually applied to a rigid rotor is inadequate to assure the rotor is balanced in its flexible mode. 

Thearle, E.L., Dynamic Balancing of Rotating Machinery in the Field, Trans. ASME Vol. 56, pp. 745-753, 1934. 

By dynamically balancing most parts, seal life and bearing life is greatly improved, even on smaller equipment. Dynamic balancing is needed on pump impellers, as the practice of static balance is woefully inadequate. Vertical pumps must be dynamically balanced the long, slender shafts are highly susceptible to any unbalanced-induced vibration. 

Larger compressors, normally above 150 to 200 horsepower, have forged steel crankshafts. Cast crankshafts are used in medium-size machines. Crankshafts should have removable balance weights to compensate for rotary unbalance as well as reciprocating unbalance. The crankshaft should be dynamically balanced when above 800 rpm. 

Rotor speeds are such that dynamic balancing is required for proper vibration control. Also, while the critical speeds are generally above the operating speed, review of the rotor dynamics should not be ignored, jwi ticularly for the dry type. 

A bubble cap tray must operate in dynamic balance, and the closer all conditions are to optimum, the better the performance for a given capacity. Evaluation of perfor-... 

Radial vaned impellers are investment cast to ensure precise contours and dynamically balanced for a smooth operation. 

Wheels are statically balanced individually then the total assembly of wheels on the shaft is also statically and dynamically balanced. An internal balancing drum (see Figure 12-40E) is usually placed on the discharge end of the shaft to balance thrust loads. 

In addition to the standard types of V belts (see Figure 3-19), many manufacturers make V belts that arc specifically designed to have a higher horsepower capacity. 4 hese V belts are shown and compared to standard V-belt cross-sections. These superpower V belts allow shorter center distances and narrow er sheaves without imposing any extra total bearing stresses. Such belts are able to reduce the drive by 30-. )0% and less for horsepow er capacity. In addition, speeds can be increased up to 6000 ft/ min without dynamic balancing of the sheaves because the sheaves arc smaller. 

The first consideration of dynamic balancing is the number of imbalance points on the rotor, as there can be more than one point of imbalance within a rotor assembly. This is especially true in rotor assemblies with more than one rotating element, such as a three-rotor fan or multi-stage pump. 

The amplitude of each point of imbalance must be known to resolve dynamic balance problems. Most dynamic balancing machines or in situ balancing instruments are able to isolate and define the specific amount of imbalance at each point on the rotor. 

Rotor speed is the final factor that must be considered. Most rotating elements are balanced at their normal running speed or over their normal speed range. As a result, they may be out of balance at some speeds that are not included in the balancing solution. As an example, the wheel and tires on your car are dynamically balanced for speeds ranging from zero to the maximum expected speed (i.e., 80 miles per hour). At speeds above 80 miles per hour, they may be out of balance. 

In order to calculate imbalance units, simply multiply the amount of imbalance by the radius at which it is acting. In other words, one ounce of imbalance at a one-inch radius will result in one oz-in of imbalance. Five ounces at one-half inch radius results in 2 oz-ins of imbalance. (Dynamic imbalance units are measured in ounce-inches, oz-in, or gram-millimeters, g-mm.) Although this refers to a single plane, dynamic balancing is performed in at least two separate planes. Therefore, the tolerance is usually given in single-plane units for each plane of correction. 

All manufacturers now publish sound pressure levels for their products and such figures should be scrutinized and compared as part of a fan-selection decision. Fans are statically, and sometimes dynamically, balanced by the manufacturer. If it is necessary to dismantle a fan for transport, it should be rebalanced on commissioning, imposing a load close to that ultimately required. 

Chemists picture equilibrium as a dynamic balance between opposing reactions. An understanding of the Law of Chemical Equilibrium can be built upon this basis. 

We have learned much about equilibrium. It is characterized by constancy of macroscopic properties but with molecular processes continuing in a state of dynamic balance. At equilibrium we can conclude that every reaction that takes place does so at the same reaction rate as its reverse reaction. 

The simple form of the equilibrium expression (4) follows directly from the dynamic nature of the solubility equilibrium. There must be a dynamic balance between the rate that iodine molecules leave the ciystal and the rate that iodine molecules return to the crystal. To understand this dynamic balance, we must consider the factors that determine these two rates. 

Thus, by expressing the dynamic balance between the rates of dissolving and precipitation, we obtain (4). The concentration of I2 at equilibrium is a constant, fixed by the temperature. This constant equals the solubility. 

In Chapter 10 we used the principles of equilibrium to help us understand solubility in liquids. In such a system constituents in solution reach the dynamic balance of equilibrium with another phase, a solid or a gas. Equilibrium can also exist among two or more constituents present in the same solution. One of the examples already encountered (in Chapter 9 and in Experiment IS) is... 

Reactions 1 to 4 are known collectively as the Chapman mechanism (first outlined by Sidney Chapman (1) in 1930. They basically explain how ozone can exist in the stratosphere in a dynamic balance it is continuously being produced by the action of solar ultraviolet radiation on oxygen molecules and destroyed by several natural chemical processes in the atmosphere. 

Muscle contraction is a delicate dynamic balance of the attachment and detachment of myosin heads to F-actin, subject to fine regulation via the nervous system. 

Here the rate of accumulation term represents the rate of change in the total mass of the system, with respect to time, and at steady state this is equal to zero. Thus the steady-state mass balance is seen to be a simplification of the more general dynamic balance. 

This, like the other dynamic balances for the CSTR, follows the full generalised form, of Sec. 1.2.5, giving... 

The coefficients of the above equations are the partial differentials of the two dynamic balance equations and are given by... 

The steady-state approach, however, provides no information on the initial transient conditions, whereby the extractor achieves eventual steady state or on its dynamic response to disturbances. For this it is necessary to derive the dynamic balance equations for the system. 

A dynamic balance for the dispersed phase holdup in stage n gives... 

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