ZAMM · Z. Angew. Math. Mech. 85, No. 3, 211 – 212 (2005) / DOI 10.1002/zamm.200310169

Short Communication

Nonlinear random vibrations of a rotating shaft

M. F. Dimentberg∗

Mechanical Engineering Department, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609, USA

Received 28 July 2003, accepted 18 May 2004Published online 8 March 2005

Key words random excitation, rotating shaft, Fokker-Planck-Kolmogorov equation, instabilityMSC (2000) 74H50

Bending vibrations of a rotating shaft due to external random excitation are considered for the case of potential instability ofthe shaft’s linear model due to presence of internal or “rotating” damping. A two-degree-of-freedom model with nonlinearelastic restoring forces is studied.An explicit expression is obtained for a stationary joint probability density of displacementsand velocities as an exact analytical solution to the corresponding Fokker-Planck-Kolmogorov equation. The results areused to develop criterion for on-line detection of instability for the operating shaft from its measured response.

c© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Consider a simple rotor consisting of a weightless shaft with disk of mass m at its midspan, rotating with angular velocityν. The shaft’s bending stiffness at infinitesimally small lateral deflections is denoted by K, whereas at moderately highdeflections it is a nonlinear function of a radial displacement R, with restoring force being always directed towards theundeformed position of a neutral axis. Furthermore, the rotor possesses external or “non-rotating” damping and internal or“rotating” damping with corresponding damping factors cn and cr respectively. Let X(t) and Y (t) be lateral horizontal andvertical displacements respectively of the horizontal shaft in the inertial frame. Then, neglecting gravity force for sufficientlyhigh rotation speeds and adding lateral random excitations the equations of lateral motion may be written as [1, 2]

X + 2κX + Ω2 [1 + f (R)] X + 2βνY = ςX (t) ,

Y + 2κY + Ω2 [1 + f (R)] Y − 2βνX = ςY (t) , where R =√

X2 + Y 2 .(1)

Here Ω2 = K/m, κ = α + β, α = cn/2m, β = cr/2m. The random forces in the RHSs of the eqs. (1) are assumed to bestationary zero-mean uncorrelated Gaussian white noises with the same intensity factor σ2. As shown in [3], this would bethe case if the excitation is produced by a turbulent flow within a machine, such as large steam turbine, with time-variantpart of the pressure field in the flow being delta-correlated both in time and in circumferential direction. Note that thecorresponding homogeneous system, i.e. one without random excitation, has an obvious solution X ≡ Y ≡ 0 which isstable if ν < ν∗ and unstable otherwise, where ν∗ = κΩ/β = Ω · (1 + cn/cr) [1, 2, 4].

The stochastic problem (1) had been solved in [3] by the method of moments [5] for the linear case, where f(R) = 0.For the present nonlinear case the theory of Markov processes can be applied [5, 6] to find a joint stationary probabilitydensity p(x, x, y, y) of the state variables X, X, Y, Y (lower-case letter as used for arguments of p correspond to the randomprocesses denoted by the same upper-case letters). To this end the eqs. (1) are rewritten as

X1 = X2 , X2 = −2κX2 − Ω2 [1 + f (R)] X1 − 2βνY1 + ςX (t) ,

Y1 = Y2 , Y2 = −2κY2 − Ω2 [1 + f (R)] Y1 + 2βνX1 + ςY (t) .(2)

The corresponding function p(x1, x2, y1, y2) satisfies the Fokker-Planck-Kolmogorov (FPK) equation [5, 6]

−x2 (∂p/∂x1) + (∂/∂x2)Ω2 [1 + f (r)] x1p + 2κx2p + 2βνy1p

− y2 (∂p/∂y1)

+ (∂/∂y2)Ω2 [1 + f (r)] y1p + 2κy2p − 2βνx1p

+

(σ2/2

) (∂2p/∂x2

2 + ∂2p/∂y22)

= 0 (3)

∗ e-mail: diment@wpi.edu

c© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

212 Short Communication

where r =√

x21 + y2

1 . Direct substitution shows that the partial differential equation (3) has the following exact analyti-cal solution

p (x1, x2, y1, y2) = C exp[− (

4κ/σ2) H +(4βν/σ2) (x1y2 − x2y1)

],

H =(Ω2/2

) (x2

1 + y21 + F (r)

)+ (1/2)

(x2

2 + y22)

, where dF/dz = f and z = r2 .(4)

Here C is a normalization constant which is reciprocal of integral of the expression (4) over infinite four-dimensional domain.The solution of the form (4) is well known for the case of the symmetric apparent nonlinear stiffness matrix in the originalequation of motion (1), that is for ν = 0 [5, 6].

The present solution will be analyzed here for the case of cubic nonlinearity, which may be result of axial tension thatappears at moderately high lateral deflections of the shaft. Thus, let

f (r) = γr2 = γz, F (r) = γz2/2= γr4/2 , where γ > 0 . (5)

By introducing scaled displacements X1 = ΩX1, Y1 = ΩY1 the solution (4) may be rewritten as

p (x1, x2, y1, y2) = (C/Ω2) exp

− (2κ/σ2)

[x2

1 + y21 + x2

2 + y22 − 2µ (x1y2 − x2y1) +

(γ/2Ω2) (

x21 + y2

1)2

](6)

where µ = ν/ν∗ = βν/κΩ. Another set of transformations may be applied next, namely U± = X1 ± Y2, V± = Y1 ± X2,resulting in the following transformed probability density

p (u+, u−, v+, v−)

= C ′ exp

− (κ/σ2)

[(u2

+ + v2−

)(1 − µ) +

(u2

− + v2+)(1 + µ) +

(γ/2Ω2)

((u+ + u−)2 + (v+ + v−)2

)2]

(7)

Here C ′ is a new normalization constant; its value is not of importance here as long as it is known to be finite.Certain important implications of the above analytical results can be formulated even without a detailed numerical study.

First of all, in the special case of a linear system (γ = 0) the stationary probability densities (4), (6), and (7) do exist indeedonly if µ < 1, that is, provided that the shaft is stable. (And they are Gaussian in this case, which had been studied in [3] bythe method of moments). If µ > 1 then the normalization integral diverges (this is especially clear from the expression (7)),so that the stationary probability density of the response does not exist, as should be expected for the unstable linear system.

In general case (γ > 0) the normalization integral is finite because of positive sign of the fourth-power terms in brackets,and therefore the stationary probability density of the (steady-state) response does exist indeed. Thus, stiffening nonlinearityin the restoring forces does restrict growth of the response level of the shaft, which is unstable in the linear approximation(as could be expected from well-known results for the shaft without random excitation [1, 2]).

Consider now qualitative changes in the response probability densities that accompany transitions between stable andunstable states of the shaft’s linear model. As can be seen from expression (4), origin x1 = x2 = y1 = y2 = 0 is a point ofmaximum of p(x1, x2, y1, y2) if µ < 1 and a saddle point if µ > 1. Furthermore, in the immediate vicinity of the origin, wherefourth-power terms are negligible, joint probability density function () of the transformed variables U± = X1 ± Y2, V± =Y1 ± X2is seen to be reduced to product of the four one-dimensional probabilities. Thus, if on-line response measurementsare made for an operating shaft resulting in four displacement and velocity signals X1(t), X2(t), Y1(t), Y2(t), the followingcriterion may be suggested for discrimination between stable and unstable conditions - that is between externally excitedoscillations and self-excited oscillations. Probability densities of the transformed measured signals U+(t) = ΩX1(t)+Y2(t)and V−(t) = ΩY1(t)−X2(t) are calculated as pU+(u+), pV −(v−). If each of these two functions is found to have maximumat zero value of its argument, the shaft is stable in the linear approximation (µ < 1) and therefore the observed response issolely due to external random excitation. If the functions are found to have minima at zero values of their arguments the shaftis unstable in the linear approximation (µ > 1) and the observed response should be qualified as self-excited oscillationsmodified due to presence of a random excitation. The criterion may be used for on-line monitoring of rotating machinerywith potential of a “mild” instability that does not lead to an immediate failure but should be timely detected.

References

[1] A. Tondl, Some Problems of Self-excited Vibrations of Rotors, Monographs and Memoranda Series, SVUSS Bechovice, Czechoslo-vakia, Bechovice (1974).

[2] G. Genta, Vibration of Structures and Machines (Springer-Verlag, New York, 1995).[3] M. Dimentberg, B. Ryzhik, and L. Sperling, J. Sound Vib. 279, 275–284 (2005).[4] Chong-Won Lee, Vibration Analysis of Rotors (Dordrecht, Kluwer, 1993).[5] M. Dimentberg, Statistical Dynamics of Nonlinear and Time-Varying Systems (Research Studies Press, Taunton, 1989).[6] Y. K. Lin and G. Cai, Probabilistic Structural Dynamics (McGraw-Hill, New York, 1995).

c© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim