The first critical case

In view of this condition, the equilibrium states associated with the first critical case are also called degenerate. Since the implicit function theorem may no longer be applied here, the persistence of such an equilibrium state in a neighboring system is not necessarily guaranteed. Thus, a transition through the stability boundary in the first critical case may result in the disappearance of the equilibrium state. 

Chap. 10. The Behavior of Dynamical Systems on. .. 10.2. The first critical case... 

Expansion strains may be taken up in three ways by bending, by torsion, or by axial compression. In the first two cases maximum stress occurs at the extreme fibers of the cross section at the critical location. In the third case the entire cross-sectional area over the entire length is for practical purposes equally stressed. 

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. 

In most cases, running speed is the forcing function that excites the natural frequency of the dynamic component. As a result, rotating equipment is designed to operate at primary rotor speeds that do not coincide with the rotor assembly s natural frequencies. Most low- to moderate-speed machines are designed to operate below the first critical speed of the rotor assembly. 

In the case of electrolytes that are capable of causing the reversal of surface charge of dispersed particles, coagulation occurs only at electrolyte concentrations corresponding to rather small values of the -potential. As electrolyte concentration becomes higher, the C, -potential decreases coagulation starts as soon as the C, -potential drops below the critical value. This corresponds to the first critical coagulation concentration, C.C.C. -1, (c, point in the curves shown in Fig. VIII-19). Further increase in electrolyte... 

The first safety cases prepared for the process industries were those developed for North Sea offshore oil and gas operations following the Piper Alpha disaster that occurred in the year 1988. The Cullen report (Cullen, 1990) that was written following that accident was highly critical of offshore operating practices and recommended that a safety case approach be implemented. The Seveso incident that occurred in Italy further prompted the increased use of safety cases. Since that time the use of safety cases has spread to other industries (such as mining and railway operations) and to other nations, primarily in Europe and Australasia. (It is notable that the safety case regime approach has not been taken up for offshore oil and gas operations in the Gulf of Mexico—instead a more prescriptive approach based on industry consensus standards is used.)... 

In the case of the first critical point the amplitudes A and B are identically equal to zero, which corresponds to the uniformly propagating wave in the original problem. The second and third critical points correspond to waves traveling along the front, which are right- and left-traveling waves, respectively. The last critical point... 

The first and second critical velocities determine the amount of material washed away the third and fourth critical velocities determine the quantity of accumulated soil. For a sandy soil, the first critical velocity is 18-22 cm/sec for a particle diameter of 0.015-0.033 cm. In this case the ratio of the second critical velocity to the first is... 

The shaft must not operate at speeds higher then 70% of the first critical speed if the impeller is not dynamic y balanced, or higher than 85% of the first critical speed if the impeller is dynamically balanced. The deflection of the shaft should be limited, particularly in the case of anchor agitators, so that the blades do not hit the walls or baffles. 

The minimum adhesion of particles to the surface will occur over a certain range of velocities. The lowest velocity value corresponds to the case in which the particles are unable to overcome the elastic properties of the surface (first critical velocity), and the upper value occurs when the particles do overcome the elastic properties of the surface (second critical velocity). Thus, adhesion of the particles occurs when the particle velocity is lower than the first or higher than the second critical velocity. The first critical velocity may be calculated if we know the elastic properties of the surfaces in contact. These calculations have been verified experimentally (see 26). The second critical velocity is only... 

The first and second critical velocities of the water and air flows as shown as functions of particle diameter in Fig. XI.7. As we should expect, the first critical velocity for wind erosion exceeds the velocity for erosion by a flow of water, owing to the difference between the adhesive forces in air and water. The broken line shows the change in the value of when the force of interaction between the particles exceeds their weight (see Fig. XI.5 and 31, 37, 38). In this case the first critical velocity produces the detachment of the adhering particles. The value of this velocity may exceed Vj 2> velocity required to produce actual flight... 

Gibbs noted that Eq. (1) may, in some Instances, be indeterminate, while Eq. (5) holds true without exception. Eq. (1) is a special case of Eq. (43) while Eq. (5) is a special case of Eq. (29). The difference between the two criteria can be seen by comparing Eqs. (27) and (42). The coefficient of the last differential term in each of the equations defines the first criticality criterion. They are equivalent i.e.,... 

There are three steps that seem to be critical. The first critical step is the selection of the mechanisms (company 3,4,7,9,11,12,13,19,20), because it is not easy to make a distinction between mechanisms enabling and mechanisms disabling a change in the behaviour of the workers (company 4) or because it is not easy to connect mechanisms with corresponding behaviours (company 9). In other cases (company 3 and 7) practitioners were not able to explain specifically the criticality of the step. 

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