system. The frequency derivatives can be advantageously used to guide the modification. Table 1 fffffffffffffffff index.
Paper No. 76–WA/DE–18
System Description For any single–branch system, an equivalent single shaft system may be determined. The equations of motion for such a system are
Sensitivity of Torsional Natural Frequencies
J• + K = 0
S. Doughty1
(1)
where J is an n n diagonal matrix of inertia values and K is an n n sti¤ness matrix,
Consulting Engineer Analytical Mechanics Company
2
Houston, Tex.
k1 6 k1 6 6 K=6 6 0 6 4 0 0
Mem. ASME Introduction Torsional vibration analysis is a common requirement in the design of engine generator sets and other rotating mechanical systems. For such systems, which are close coupled and lightly damped, the Holzer method of analysis is widely applied [1]2 . If a natural frequency is found to coincide with an external forcing frequency, it is usually required that the system be detuned. To minimize the design changes required, alternate couplings of di¤erent sti¤nesses are often the preferred remedy. If satisfactory detuning cannot be achieved by varying the couplings, changes are next considered for the moments of inertia, usually increases to such items as the ‡ywheel or generator fan. In too many cases, this becomes an expensive, tedious search conducted with little theoretical guidance. A sensitivity analysis of the natural frequencies with respect to the design parameters is needed.
k1 k1 + k2 k2
0 k2 k2 + k3 .. .
0 0
0 0 .. . ..
0
3
0 0 0
. kn
1
7 7 7 7 7 7 5
(2)
If a sinusoidal solution is assumed, the equation of motion reduces to the ordinary eigenproblem !2 J + K = 0
(3)
The solutions of equation (3) include ! = 0 since the system is free–free. This is a rigid body mode and is of no interest for the analysis of vibrations. The Holzer process is an iterative search for the eigenvalues as roots of the characteristic equation and also yields the associated eigenvectors de…ned by equation (3). In practice, for large systems, only the lower mode frequencies and associated eigenvectors are obtained. For present purposes, it is assumed that a mode of interest has been identi…ed and that the eigenvector has been normalized such that
The derivatives of the eigenvalues and eigenvectors have been explored by others, reported largely in the aerospace literature [2–5]. The purpose of the present paper is to apply these methods to the torsional vibration problem to obtain a sensitivity analysis for the natural frequencies. Assuming that the Holzer analysis for a particular mode is available, the sensitivity calculations for that natural frequency are su¢ ciently simple to permit manual computation, or they are readily adapted to computer implementation within a Holzer analysis program.
T
J =1
(4)
Natural Frequency Derivatives Following the method used by Rogers [2], the Rayleigh quotient is …rst di¤erentiated with respect to an unspeci…ed parameter, v. The Rayleigh quotient is the product of the equation of motion for a particular mode (eq. (3)) with the mode vector for that same mode,
1 Presently Assoc. Professor, Department of Mechanical Engineering, Lousiana Tech University, Ruston, LA. 2 Numbers in brackets designate References at end of paper. Contributed by the Vibration and Sound Committee of the Design Engineering Division for presentation at the Winter Annual Meeting, New York, December 5–10, 1976, of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS. Manuscript received at ASME Headquarters August 2, 1976. Paper No. 76–WA/DE–18. Also published in Journal of Engineering for Industry, Trans. ASME, Ser. B, Vol. 99, No. 1, Feb. 1977, pp. 142–143.
!2
T
J +
T
K =0
(5)
After di¤erentiating this with respect to the parameter v, and using the equation of motion (3) and the normalization condition (4) to reduce the result, the frequency derivative is 1
@! = 0:5! @v
1 T
@K @v
0:5!
T
@J @v
Table 1 ————————————————— index Ji ki i kg-m2 N-m/rad 1 cylinder 0:10 1:00 105 2 cylinder 0:10 0:50 105 3 ‡ywheel 0:50 0:25 105 4 fan 0:30 1:00 105 5 gen rotor 1:10
(6)
For the torsional vibration problem, the interesting candidates for v are the inertias and the sti¤nesses. When v is taken to be one of the inertias, Js , the @K=@Js vanishes and @J=@Js is identically zero except for a +1 in the (s; s) position. After multiplying out the products, the result is
@! = @Js
0:5!
2 s
Table 2 ! = 561 rad/s —————————————————— @! @! index Eigen– @Ji @ki i vector
(7)
i
1 2 3 4 5
For the derivative with respect to a sti¤ness, ks , the @J=@ks vanishes and @K=@ks consists of four nonzero elements: there is a +1 in the (s; s) and (s + 1; s + 1) positions and a 1 in the (s; s + 1) and (s + 1; s) positions. The result of the matrix product is
@! = 0:5! @ks
2
1 s
s+1
+2:104 +1:442 0:790 0:278 +0:113
rad kg-m 2 -sec
0:124 0:583 0:175 0:216 0:357
4
10 103 103 102 101
rad N-m -sec
0:391 0:444 0:234 0:136
10 10 10 10
3 2 3 3
References (8) 1. Holzer, H., Die Berechnung der Drehschwingungen, Springer, Berlin, 1922.
Equations (7) and (8) are the sensitivity derivatives which were sought. Note that the derivative of frequency requires only the frequency value and the elements of the associated eigenvector. A typical application of these results is considered in the following example problem.
2. Rogers, L.C., “Derivatives of Eigenvalues and Eigenvectors,” AIAA Journal, Vol. 8, No. 5, May 1970, pp. 943–944. 3. Garg, S., “Derivatives of Eigensolutions for a General Matrix,” AIAA Journal, Vol. 11, No. 8, Aug.. 1973, pp. 1191–1194.
Example Problem
4. Farshad, M. “Variations of Eigenevalues and Eigenfunctions in Continuum Mechanics,” AIAA Journal, Vol. 12, No. 4, Apr. 1974, pp. 560–561.
The data listed in Table 1 may be considered to represent a two cylinder engine (J1 ; J2 ) with a ‡ywheel (J3 ) driving a generator consisting of a fan (J4 ) and rotor (J5 ). The results of a Holzer analysis for the second mode of the system are given in Table 2. The eigenvector (second column) is normalized according to equation (4). From the last column, the system is seen to be relatively insensitive to the coupling sti¤ness, k3 , requiring almost an order of magnitude increase to achieve a 10 percent increase in frequency. Note also that the system is much more sensitive to a change in the ‡ywheel inertia (J3 ) compared to a change in the fan inertia (J4 ). The decision as to how to accomplish a particular frequency shift must also consider the impact of a proposed change on the other modes of the system. The frequency derivatives can be advantageously used to guide the modi…cation.
5. Rudisill, C.S., “Derivatives of Eigenvalues and Eigenvectors for a General Matrix,” AIAA Journal, Vol. 12, No. 5, May 1974, pp. 721–722.
2
