Sensitivity of Torsional Natural Frequencies for a Branched System: A Three Stage Centrifugal Compressor

S. Doughty Assistant Professor

Department of Mechanical Engineering Texas A&M University College Station, Texas 77843 October, 1980 1

Nomenclature ki = torsional sti¤ness of ith shaft section ni = number of teeth on gear at station i fAg = any eigenvector, also used with a subscript to refer to a speci…c eigenvector Ji = polar mass moment of inertia at station i [M ] = diagonal generalized inertia matrix [S] = generalized sti¤ness matrix = physical rotation coordinate for station i i = generalized coordinate i ! i = natural frequency for ith mode Introduction In all rotating machinery trains there exists the potential for destructive torsional vibration. During system design, the natural frequencies are customarily calculated to assure that they are well removed from all expected torsional excitation frequencies. If a natural frequency is found too close to an excitation frequency, it is necessary to detune the system. For detuning, it is important to identify those parameters that substantially a¤ect the natural frequency to be shifted. The e¤ect of these same parameters on the other natural frequencies is also needed to avoid simply exchanging one di¢ culty for another. All of this information is determined in a comprehensive sensitivity analysis of the natural frequencies with respect to the system inertia and sti¤ness parameters. For systems reducible to an equivalent single shaft system, the required analysis has been previously established [1]. A similar analysis for branched systems is the subject of the present paper. For single shaft systems the form of the sti¤ness matrix is …xed, and consequently, the general case is readily analyzed. There is no limit to the variety of branched systems and their associated sti¤ness matrices, so that no general case exists. Rather, the method of natural frequency sensitivity analysis will be presented by reference to a particular branched system: a motor driven, three stage centrifugal compressor of the sort commonly used to supply plant air systems. The method is illustrated by a numerical example for such a compressor. System Description The centrifugal compressor to be considered here is represented by the mass–elastic diagram shown in Fig. 1. The system consists of an electric motor driving a bull gear which in turn drives the two high speed shafts carrying the impellers for the three stages. Physical coordinates, i , and polar mass moments of inertia, Ji , i = 1; :::; 7 are indicated for each rotating mass. The torsional sti¤nesses, ki , i = 1; 2; 3; 4 provide 2

elastic connections between the various components. The numbers of teeth on the bull gear and the pinions are n2 ; n3 ; and n5 ; respectively.

Fig. 1: Mass –elastic diagram for motor driven centrifugal compressor, a typical branched torsional system. Neglecting any ‡exibility in the gears, the rotations of the bull gear and the two pinions are kinematically related

n2 n2

2 2

= n3 = n5

3 5

Since there are then only …ve degrees of freedom, a suitable set of generalized coordinates are the …ve ’s de…ned as follows:

3

2

= =

3

=

4

=

5

=

6

=

7

=

1

1 2

n2 n3 n2 n3 n2 n5 n2 n5 n2 n5

2

3

2

4

5

In terms of the generalized coordinates, the equations of motion for the free vibrations of the system are n o [M ] • + [S] f g = f0g

where

[M ] = diag J1 ; J2 + J3

2

n2 n3

+ J5

n2 n5

2

2

n2 n3

; J4

n2 n5

; J6

2

; J7

2

n2 n5

!

and 2

6 6 6 6 6 6 [S] = 6 6 6 6 6 6 4

k1 k1

k1

0

+ (k3 + k4 ) 0 0 0

k2 k3 k4

n2 n3 n2 n5 n2 n5

0

2

n2 n3

k 1 + k2

0

n2 n5 2

n2 n3

k2

2

k2

2

n2 n3

n2 n5

k3

n2 n5

k4

0 k3

n2 n5

0 2

0

2

0

2

2

2

0

2

0

k4

n2 n5

2

3 7 7 7 7 7 7 7 7 7 7 7 7 5

With a sinusoidal time dependence of the form fAg ej!t assumed, the equations of motion are transformed into the algebraic eigenproblem 4

! 2 [M ] fAg + [S] fAg = f0g The solutions for the eigenproblem may be obtained by any of several methods; in the numerical example given later the Jacobi method is used [2]. The complete set of solutions will always include a zero frequency mode that is inherent in the unrestrained nature of rotating machine trains. The zero frequency mode is disregarded throughout the remainder of this discussion. For all other modes of interest it is assumed that an eigensolution has been determined, and that each mode vector has been normalized such as to satisfy fAi gT [M ] fAi g = 1:0 Natural Frequency Derivatives For any mode of free vibration, the natural frequency and the mode vector satisfy the de…ning eigenproblem. The Rayleigh quotient for a mode is a scalar, formed by premultiplying the eigenproblem by the transpose of the mode vector. For the ith mode, the Rayleigh quotient is ! 2i fAi gT [M ] fAi g + fAi gT [S] fAi g = 0 The natural frequency derivatives are obtained by di¤erentiating the Rayleigh quotient with respect to the individual system parameters. Consider …rst the derivative with respect to a particular mass moment of inertia, Jp . Di¤erentiating the Rayleigh quotient with respect to this parameter, and recognizing that all of the inertia dependence is in [M ] gives @! i = @Jp

@ 1 ! i fAi gT [M ] fAi g 2 @Jp

The derivative of the inertia matrix with respect to Jp will contain only one nonzero element, +1 or a tooth ratio squared, appearing on the diagonal as seen from the form of the expression for [M ]. As an example, @ [M ] =@J3 is

@ [M ] = @J3

n2 n3

2

2 6 6 6 6 4

5

0 0 0 0 0

0 1 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

3 7 7 7 7 5

Thus the derivative of a natural frequency with respect to an inertia consists of the following factors: negative one half the natural frequency, the square of a mode vector components, and in some cases the square of a tooth ratio. This the same as for a single shaft system except for the tooth ratio factor. For the derivative with respect to an individual sti¤ness, the Rayleigh quotient is di¤erentiated recognizing that all the sti¤ness dependence is in the sti¤ness matrix, [S], 1 @ @! i = fAi gT [S] fAi g @kp 2! i @kp The derivative of the sti¤ness matrix, @ [S] =@kp , is entirely zero except for four elements, each with the same value except for sign, a +1 or a tooth ratio squared (see the form for the sti¤ness matrix). A typical derivative matrix is

@ [S] = @k3

n2 n5

2

2 6 6 6 6 4

0 0 0 0 0 0 +1 0 1 0 0 0 0 0 0 0 1 0 +1 0 0 0 0 0 0

3 7 7 7 7 5

The frequency sensitivity with respect to a shaft sti¤ness includes as factors one half the reciprocal of the frequency involved, the squared di¤erence of two components of the mode vector, and perhaps a tooth ratio squared. This is somewhat similar to the single shaft case except for the tooth ratio factor and the choice of mode vector components involved in the di¤erence. For single shaft systems the di¤erence always involves adjacent mode vector components. As demonstrated here, the di¤erence may involve more widely separated elements of the mode vector when branched systems are considered. In each case, it is a re‡ection of the system connectivity as described by the station numbering scheme employed. The computations just described provide the derivatives of one natural frequency with respect to each of the system design parameters. They are readily repeated to give derivatives for each mode of interest, provided that the eigensolution is known for each such mode. Example Numerical values for the design parameters of the three stage centrifugal compressor are given in Table 1. Numerical solution of the eigenproblem using these values shows the …rst and second twisting mode frequencies at 292:0 and 3672:6 rad/sec with mode vectors as listed in Table 2. The results of sensitivity calculations for these mode are 6

shown in Table 3. The derivatives of frequency with respect to moment of inertia is negative in all cases while the derivatives with respect to sti¤ness are positive in all cases (as expected). The second stage components, J6 and k3 , have a large e¤ect on both frequencies as shown in Table 3. If it is desired to move both ! 1 and ! 2 in the same sense, a change in one of these parameters may be a good choice. On the other hand, if it is necessary to change ! 1 with little e¤ect on ! 2 , modi…cations to J1 or k1 are probably in order. Before making such changes it is necessary to check other aspects of the design that may be adversely a¤ected. Table 1 Three Stage Centrifugal Compressor Parameters Index, i Ji ki ni 2 (kg–m ) (N-m/rad) 1 9:372 6:310 105 1 2 2:956 10 2:958 105 349 3 4 3 6:246 10 9:359 10 38 2 4 4 1:746 10 5:280 10 5 3:904 10 4 19 3 6 7:349 10 7 3:036 10 3 Table 2 Eigensolutions !1 =

292 rad/s

+2:903 7:723 fA1 g = 7:762 7:775 7:761

!2 = 1

10 10 10 10 10

2 2 2 2

3672:6 rad/s

+1:778 3:544 fA2 g = 1:738 +5:998 1:579

10 10 10 10 10

4 2 1 1 1

Table 3 Frequency Sensitivity Coe¢ cients Index, i 1 2 3 4 5 6 7

@! 1 @Ji 2

1

kg-m -s 1:230 101 8:708 10 1 7:345 101 7:420 101 2:938 102 2:978 102 2:967 102

@! 1 @ki

(N-m-s) 1 2:313 10 4 2:204 10 8 1:565 10 7 8:362 10 8

@! 2 @Ji 2

1

kg-m -s 5:806 10 5 2:307 1:946 102 4:681 103 7:783 102 2:229 105 1:545 104 7

@! 2 @ki

(N-m-s) 1 1:728 10 7 2:200 10 4 1:854 10 2 6:889 10 4

Conclusion It has been shown that the frequency sensitivity calculation method previously established for a single shaft system may be readily adapted to branched systems. As before, a knowledge of the eigensolutions is required as a basis for the sensitivity calculation. The details of the sensitivity calculation are directly tied to the form of the sti¤ness matrix which in turn depends on the system con…guration. For any …xed con…guration, the sensitivity calculation is readily adapted to computer evaluation. References

1. Doughty, S. “Sensitivity of Torsional Natural Frequencies,” Journal of Engineering for Industry, Trans ASME Ser. B, Vol. 99, No. 1, Feb. 1977, pp. 142–143. 2. Carnahan, B., Luther, H.A., and Wilkes, J.O., Applied Numerical Methods, 1st ed., Wiley, New York, 1969, pp. 250–260.

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