Mode shape

To summarize the importance of the critical speed concept, one should bear in mind that it allows an identification of the operation region of the rotor-bearing system, probable mode shapes, and approximate locations of peak amplitudes. 

Initially, tests are run to identify the major eritieal frequeneies of the impeller. Mode shapes are then determined visually at eaeh of the eritieal frequeneies. To obtain these mode visualizations, salt is sprinkled evenly on the dise surfaee. The shaker is maintained at a partieular frequeney, at whieh value a given eritieal frequeney is exeited for a eertain length of time so that the salt partieles display the mode shape. The salt aeeumulates in the nodal regions. Photographs are taken at lower values of these eritieal frequeneies. Photography allows a qualitative identifieation of the appropriate mode shapes eorresponding to eaeh frequeney. Figure 5-27 shows an impeller with the mode shapes. 

The actual field rotor operates with characteristic mode shapes significantly different than those that occur during a standard production balance. 

The mode shapes (problem unbalance distributions) of the rotor at the criticals. 

Figure 9-9. Typical mode shapes of an undamped system.

The solutions to a problem of this magnitude can be found in references [3, 7] and others. Figures 9-16 and 9-17 are torsional mode shape diagrams of some typical systems. While the rigorous solution to the multimass damped system is not within the scope of this book, several interesting points should be made. 

Meiidional tiow vector, 152 Meter run requirements, 431 Method of piping, 190 Michell bearing, 201 Mini lube system, leeiprocating, 78 Mixed flow, II, 14 Mixed-flow impeller, 14 Mixture compressibility, 20 Mode shape,s, 386, 388 Moisture corrections, 2 Mole, U ... 

Although the concern is primarily for the response of the piping. system, the possibiliis of dynamic coupling with the containment structure should not be neglected. A concern is whether or not the secondary shield wall will withstand the dynamic interaction between the walls and the pump. This is answered by examining the mode shapes if there were no coupling between the walls and the pump. 

The natural frequency, co associated with the mode shape that exhibits a large displacement of the pump is compared with the fundamental frequency, of the wall. If co is much less than ru, then the dynamic interaction between the wall and the loop may be neglected, but the kinematic constraint on the pump imposed by the lateral bracing is retained. If nearly equals nr , the wall and steam supply systems are dynamically coupled. In which case it may be sufficient to model the wall as a one-mass system such that the fundamental frequency, Wo is retained. The mathematical model of the piping systems should be capable of revealing the response to the anticipated ground motion (dominantly translational). The mathematics necessary to analyze the damped spring mass. system become quite formidable, and the reader is referred to Berkowitz (1969),... 

One of the major complications in the plate buckling solution is the need to investigate the influence of buckle mode shape on the buckling load itself. That is, the plate buckling load in Equation (5.81) is a function... 

The free vibration frequencies and mode shapes will be determined for plates with various laminations specially orthotropic, symmetric angle-ply, antisymmetric cross-ply, and antisymmetric angle-ply. The results for the different types of lamination will be compared to determine the influence of bend-twist coupling and bending-extension coupling on the vibration behavior. As with the deflection problems in Section 5.3 and the buckling problems in Section 5.4, different simply supported edge boundary conditions will be used in the several problems presented. 

Shaft stiffness Most machine-trains used in industry have flexible shafts and relatively long spans between bearing-support points. As a result, these shafts tend to flex in normal operation. Three factors determine the amount of flex and mode shape that these shafts have in normal operation shaft diameter, shaft material properties, and span length. A small-diameter shaft with a long span will obviously flex more than one with a larger diameter or shorter span. 

Most turbines have relatively long bearing spans and highly flexible shafts. These factors, coupled with variations in process flow conditions, make turbine rotors highly susceptible to shaft deflection during normal operation. Typically, turbines operate in either the second or third mode and should have narrowbands at the second (2x) and third (3x) harmonics of shaft speed to monitor for mode shape. 

Because of the length of these shafts and the flexible couplings or joints used to transmit torsional power, jackshafts tend to flex during normal operation. Flexing results in a unique vibration profile that defines its operating mode shape. 

The following parameters are monitored in a typical predictive-maintenance program for fans aerodynamic instability, running speeds, and shaft mode shape, or shaft deflection. 

A narrowband window should be established to monitor the fundamental (lx), second (2x), and third (3x) harmonic of shaft speed. With these windows, the energy associated with shaft deflection, or mode shape, can be monitored. 

There is a potential for unstable flow through pumps, which is created by both the design-flow pattern and the radial deflection caused by back-pressure in the discharge piping. Pumps tend to operate at their second-mode shape or deflection pattern. This mode of operation generates a unique vibration frequency at the second harmonic (2x) of running speed. In extreme cases, the shaft may be deflected further and operate in its third (3x) mode shape. Therefore, both of these frequencies should be monitored. 

Running-speed harmonics When setting bandwidth, at least three harmonics of running speed should be included to ensure the ability to quantify the operating-mode shape of the shaft. This is accomplished by setting Fmax to at least three times the running speed. 

Trend data can be used in the following ways (1) to compare with specific reference values, (2) mode-shape comparisons, and (3) cross-machine comparisons. 

A clear understanding of the mode shape, or shaft deflection, of a machine s rotating element is a valuable diagnostic tool. Both broadband and narrowband filtered energy windows can be used at each measurement point and orientation across the machine. The resultant plots, one in the vertical plane and one in the horizontal plane, provide an approximation of the mode shape of the complete machine and its rotating element. 

The overall energy from the filtered broadband plotted against measurement location provides an approximation of the mode shape of the installed machine. Figure 44.31 illustrates a vertical broadband plot taken from a Spencer blower. Note that the motor appears to be flexing in the vertical direction. Extremely high amplitudes are present in the motor s outboard bearing and the amplitudes decrease at subsequent measurement points across the machine. 

Figure 44.31 Vertical broadband mode shape for Spencer blower indicates potential failure...

Figure 44.32 Horizontal narrowband (lx) mode shape indicates shaft deflection...

In most cases, this failure mode also excites the third (3x) harmonic frequency and creates strong axial vibration. Depending on the severity of the instability and the design of the machine, process instability also can create a variety of shaft-mode shapes. In turn, this excites the lx, 2x, and 3x radial vibration components. 

Rather than evaluate each measurement point separately, plot the energy of each measurement point on a common shaft. First, the vertical measurements were plotted to determine the mode shape of the machine s shaft. This plot indicates that the outboard end of the motor shaft is displaced much more than the remaining shaft. This limits the machine problem to the rear of the motor. Based strictly on the overall value, the probable cause is loose motor mounts on the rear motor feet. The second step was plotting the horizontal mode shape. This plot indicates that the shaft is deflected between the pillow block bearings. Without additional information, the mode shaft suggests a bent shaft between the bearings. 

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