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Effects of Damping and Circulatory Forces on DynamicInstability of Gyroscopic Conservative ContinuousSystemsRong C. Shieh aa MRJ , INC , FAIRFAX, VIRGINIA, 22030Published online: 29 Mar 2007.

To cite this article: Rong C. Shieh (1983) Effects of Damping and Circulatory Forces on Dynamic Instability of GyroscopicConservative Continuous Systems, Journal of Structural Mechanics, 11:2, 197-213, DOI: 10.1080/03601218308907441

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J. STRUCT. MECH., 11(2), 197-213 (1983)

Effects of Damping and Circulatory Forces

on Dynamic Instability of Gyroscopic

Conservative Continuous Systems

Rong C. Shieh

MRJ, INC. FAIRFAX, VIRGINIA 22030

ABSTRACT

The title problem is studied, with emphasis on the small damping and circulatory force case. It is shown that small internal and/or external damping forces and/or small (as well as large) circulatory forces in general destabilize an otherwise stable gyroscopic conservative system. A condition for no destabilizing effects of these small forces is obtained. A concept of "perfect" system in elastic stability of nonconservative problems is also presented. An example problem is given for demonstration purposes.

I. INTRODUCTION

The study of dynamic instability of general gyroscopic conservative and nonconservative systems is not only of theoretical interest, but is also of practical interest and importance. To underscore this point, we merely need to cite two classes of practical dynamic instability problems of gyroscopic systems; namely, the critical rotating speeds of elastic shafts and critical

Copyright 1983 by Marcel Dekker, Inc. 0360-1218/83/11024197$3.50/0

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198 SHIEH

flow speeds of fluid-conveying elastic pipes. For earlier studies (prior t o mid-1950) of the dynamic instability of general gyroscopic systems, we cite the works of Thomson and Tait [5], Ziegler [6, 71, Beth [8], and Bottema [9]. These studies were mostly confined to the investigation of stabilizing or destabilizing effects of gyroscopic forces on nondissipative conservative or circulatory elastic systems. Since then, there appears to be very little activity in this area of study until the studies of Shieh and Masur [I] and Shieh [2, 31. In these studies, general asymptotic and elastic (quasi-) stability criteria of an elastic or Kelvin-Voigt solid body were formulated in an "equivalent energy" form [I], variational principles were established [2, 31, a variational method similar to that of the Raleigh-Ritz for finding critical loads was formulated [2-41, and special features of instability phenomena that are typical of various dissipative (damped) and nondissipative (undamped) systems were discussed [I, 21.

Following these studies, dynamic instability of gyroscopic conservative systcms was further investigated by Huseyin and Plaut [lo, I I] (also see Ref. 12). In Ref. 13, Shieh studied the elastic stability problem of a shaft subjected to either circulatory type torsions or combined axial dead end force and constant rotation and formulated or established stability criteria and variational principles for shafts with unequal rigidities (e.g., elliptic shafts) in higher order "equivalent" energy terms.

The present study is an extension of Refs. I and 2. Emphasis of the study is placed on destabilizing effects of small external, as well as internal, damping forces and/or small circulatory type forces and finding no destabilizing con- ditions of these forces.

The governing equations and stability conditions formulated in Ref. 1 are briefly presented in Section 11. In Section 111, effects of small damping and circulatory forces on stability of gyroscopic conservative (g.c.) systems are studied; a condition that the stability load regions of the latter are unaffected by the presence of small damping and circulatory forces is ob- tained; general destabilizing effects of these forces on a n otherwise stable g.c. system are shown to exist; a special case of the above principles, applied to conservatively loaded rotating elastic shafts, is discussed; and a concept of "perfect" system in elastic stability of nonconservative problems is pre- sented. Section IV deals with the corresponding discrete systems and also contains illustrative numerical example problems of a conservatively loaded rotating system.

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GYROSCOPIC CONSERVATIVE CONTINUOUS SYSTEMS 199

II. GOVERNING EQUATIONS AND STABILITY CONDITIONS

The brief presenration of this section generally follows that of Ref. 1. These materials are presented here for completeness.

Consider a Kelvin-Voigt type viscoelestic solid body with material volume V and material surface S. The body is originally in equilibrium (designated as State I) under the set of body forces B, (per unit undeformed volume) and surface tractions F, (per unit undeformed surface area) on the part ST of S. On the remaining part of boundary surface S-ST, the displacement @ is prescribed to be @:. In order to investigate the local stability of this equi- librium state (State I), we apply small initial disturbances a t time t = 0 t o the equilibrated solid body and study the behavior of subsequent motions in the neighborhood of State 1.

We shall assume that the applied body forces B(R, t) (R = material posi- tion vector) and surface tractions F(R, t) contain both displacement-depend- ent (conservative and nonconservative (circulatory)) components and velocity-dependent (dissipative and gyroscopic) components. In addition, surface tractions are assumed to contain acceleration (&)-dependent com- ponents of boundary inertia force type. Thus, after linearization with respect to a small incremental displacement vector 4 = @ - @, and velocity vector 4 = 6 - &,, the applied forces and tractions can be written as

B = Bl + I(4) + m(&, in V

F = F, + L(4) + ~ ( 4 ) - qd, on ST (1) = o n S - S T

where m, I, L, and M are linear operators on 4 or 4, q is the boundary (or apparent) mass per unit undeformed surface area, and a dot indicates material time (r) derivative.

The equations of small motion superimposed on a finitely deformed equi- librium state can be obtained from Eq. 12 of Ref. 1 by adding the acceleration term -q$ to the traction boundary conditions. Introducing in these equa- tions the complex response

where u, v, /?, and o are real quantities, one obtains

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where P = rV, P' = Vr) and quantities A, r, B, and Tare evaluated a t State I (obtained by solving equations governing the equilibrium State I). In Eq. 3, V = d/aR is the gradient operator, R is the material position vector, r (= R + @,) is the spatial position vector, A and B are elastic and viscous tangent moduli (at State I), T is the piola Kirchhoff stress tensor of the second kind, N is the outward unit normal vector of the undeformed surface element, and superscript t stands for "transpose."

Equations 3 are assumed to yield the set of modal solutions

Assuming that the modal expansion is valid, we define stability/instability conditions as follows:

(1) Stability: Asymptotic stability: All b, < 0 (occurs in damped systems) Quasi-stability (corresponding to a critical case in the Liapunov sense of

stability [14]): All P, # 0 ; all w, = 0 (occurs in undamped elastic systems)

(2) Instability: At least one /?, > 0 (3) Critical state:

Asymptotic critical state: At least one P, = 0; remaining /?, < 0 Quasi-critical state: All P, = 0 and a t least a pair of w, repeated (including

w, = 0 case)

Instability and critical states can be further classified as divergence o r flutter type, according to whether at least one w, = 0 or all o, f 0 among those 1, = P, + iw, with p, > 0.

Multiplication of the first of Eqs. 3 by w* = u - iv (the complex conju- gate of w), integration over V, and use of the divergence theorem and bound- ary conditions in Eqs. 3 leads to the scalar equations

where

and

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GYROSCOPIC CONSERVATIVE CONTINUOUS SYSTEMS 2Q1

[(Vw .P): i: ( P C . wtV)]dV

= A / [ ( V u . P ) : B : ( P f . u V ) + ( V V . P ) : ~ : ( P ~ . V V ) ] ~ V 2 Y

( 6 ~ )

D,= -- I I [w*. m(w) + w.m(w*)]dV - [w*.M(w) + w.M(w*)]dS 4 v

= -1 1 [U . m(u) + v . m(v)]dv - + V . M(v)]dS 2 v ( 6 4

G = A / [w*. m(w) - w - m(w*)]dV + 4; [w* .M(w) - w.M(w*)]dS 4i ,, ' Is.

- 1 / E (Vw* . wV)dY 2 v

1 N = 1 / (r* . I(.) - a l(w*)]dV + 4; [w* . L(w) - w . L(w*)]dS

4i 1.

K, D,, D,, U, C, and N are real quadratic scalar functianals of u and v ; i.e., the real and imaginary parts of eigenfunctions w = u + iv. The funclionals Kw2, D p , D,o, GO, P, and N are, respectively, the equivalent (or general- ized) kinetic energy (i.e., K i n Eq. 6b, with w* and w replaced by @* and a), internal and external viscous dissipation energies per radian in the time interval (0, 2 x / o ) , gyroscopic potential, and work per radian done by purely circulatory force component in the time interval (0, 2x10) under the motion 4 = weiW'.

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Based on the definitions of stability given earlier and Eqs. 5, the following conditions are obtained:

(1) Asymptotic stability conditions: L > 0, R > 0 (Eq. 7a); (Rw - N)/ (2Kw - G) > 0 (Eq. 7b).

(2) Quasi-stability conditions (R = 0): L = 0, N = 0 (w t 0) (Eq. 8) for all normal modes.

(3) Instability conditions: L < 0 for a t least one normal mode or alter- nately (R - N)/(2Kw - G) < 0.

(4) Asymptotic critical: L = 0, Rw - N = 0 (Eq. 9) for a t least one mode, while the remaining modes satisfy Eq. 7a.

(5) Quasi-critical (for the case of N = 0): L = 0, 2Kw - G = 0 (G $ 0) (Eq. 10) for at least one mode, while the remaining normal modes satisfy Eq. 8.

Ill. EFFECTS OF SMALL DAMPING AND CIRCULATORY FORCES ON GYROSCOPIC CONSERVATIVE SYSTEMS

We observe from Eq. 8 that a circulatory force, no matter how small, in general destabilizes an otherwise stable gyroscopic conservative system. This is because one of the stability conditions, N = 0 in Eq. 8, in general cannot be met unless all normal modes are real o i purely imaginary (but not both), which is impossible unless o = 0 under all normal modes, due to the presence of gyroscopic forces. However, if in addition to circulatory forces, finite dissipative (damping) forces are also present in a gyroscopic conservative system, the resulting system can again possess a finite region of asymptotic stability, as can be seen from Eq. 7a. If damping is small, in view of Eq. 7b, the circulatory force component, in general, is also necessarily small in order that the system can be stable. In general, small damping and circulatory forces may destabilize an otherwise stable gyroscopic conservative system. We shall defer the study of this behavior until later and first obtain the condition for identity of quasi-critical and asymptotic critical loads.

A. Identity Condition of Quasi-Critical and Asymptotic Critical Loads

We observe a similarity between quasi-critical conditions of Eq. 8 for gyroscopic conservative systems (designated as g.c. systems) and asymptotic

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GYROSCOPIC CONSERVATIVE CONTINUOUS SYSTEMS U)3

critical conditions of Eq. 9 for dissipative, gyroscopic, circulatory systems (designated as d.g.n. systems). These conditions become identical, a t least in form, if we replace 2K and G in the second equation of Eq. 10 with R and N, respectively, or if the damping and circulatory forces are such that

if

4, = klgQ (k, = const.) (1 lb)

for the asymptotic critical normal mode 4 , and quasi-critical normal modes 4Q, where p, and pQ are the loading parameters and subscripts A and Q stand for "asymptotic critical" and "quasi-critical," respectively. It should be noted that satisfaction of Eq. I l a does not necessarily imply p, = p,, in general, unless critical normal modes of both systems differ only by a constant factor, as shown in Eq. I lb.

In the case of small damping and small circulatory forces, one can write

where E is a small damping parameter and R' and N' have the same orders of magnitude as those of K and G. If these forces are vanishingly small (E -t

0 +), deviations of normal modes from those of the original g.c. system are also vanishingly small. Therefore, 4, = k l g Q and all asymptotic and quasi- critical conditions become identical if the first of Eqs. I la, which now can be written as

is satisfied for the asymptotic and quasi-critical normal modes. Note that the side condition PQ < 0 is a necessary condition for existence of a quasi- critical or flutter state, in view of the quasi-stability condition of Eq. 8 with N m 0, which can also be expressed as

Thus, if P 2 0, a g.c. system cannot be flutter critical or lose stability by flutter. The condition P = PQ = 0 corresponds to both asymptotic and quasi- critical divergence states (w = 0) and the corresponding critical loads for both g.c. and d.g.n. systems are apparently identical, if the circulatory force component is vanishingly small. Therefore, one may state the following.

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204 SHIEH

Theorem 1 : Under the influence of vanishingly small damping and cir- culatory forces, the quasi-critical flutter loads (if any) of the original g.c. system are identical to the asymptotic critical flutter loads of the resulting d.g.n. system if and only if the condition given by Eq. 13 is satisfied. The quasi- and asymptotic critical divergence loads (if any) governed by P = 0 are always identical.

As a special case, we consider the case in which internal damping forces are absent (D, = 0) or negligibly small, compared with external damping forces, and external forces are of the form

where E, and c, are the linear external viscous body force and surface trac- tion coefficient tensors, respectively, and subscripts G, C, and N astand for "gyroscopic," "conservative," and "nonconservative" (circulatory) com- ponents of the applied forces, respectively. Under these force components, Eq. 14 is apparently satisfied if

Furthermore, if the external damping forces coefficients are isotropic; i.e.,

e, = eVZ, e, = &,I (I = identity matrix) (154

Equation 1 can be written as

where k , is the proportional constant and R, is the position vector on the surface S,.

Equation 15d is a sufficient condition for quasi-critical and asymptotic critical loads to be identical.

B. Destabilizing Effects of Small Damplng andlor Small Circulatory Forces

Theorem 2 : If the condition of Eq. 13 is violated, vanishingly small damping (whether internal or external) and/or circulatory forces destabilize an otherwise stable g.c. system.

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GYROSCOPIC CONSERVATIVE CONTINUOUS SYSTEMS UH

T o show this, we rewrite the asymptotic critical condition of Eq. 7 in the following form:

H/4K = K(w - G/2K)' (1 6 4 = N/R = Nf/R' (16b)

If the condition of Eq. 13 is violated, the asymptotic critical condition of Eq. 16 implies that

H > O (164

for the asymptotic critical mode. This, in turn, implies quasi-stability of the corresponding g.c. system (cf. Eq. 14). Furthermore, it is necessary that H > 0 for both quasi-stability and asymptotic stability. Hence, the asymp- totic stability load region must be a subregion of quasi-stability.

C. Conservatively Loaded Rotating Elastic Shaft Case

In the case of a conservatively loaded rotating body with constant angular velocity vector 52, the force components corresponding to those in Eq. 15a are

and T + 0 in Eq. 6g if 6, is with respect to the rotating coordinate system. If we assume that the external damping forces are of viscous traction force type and are proportional to the absolute velocity vector 4, = 8, t R x 4, then

where L,(4) and e, are defined as in Eq. 15a and subscript D stands for the damping component of velocity-dependent forces. Strictly, small external as well as internal damping forces in general destabilize a conservatively loaded, rotating elastic body because the conditions of Eqs. 13 or 15b or 15d in general cannot be satisfied. If, however, one makes the same approxima- tions as elementary beam theory, Eqs. 17 a and b can be integrated over the cross-sectional area A and circumferential length s to yield

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where z is the coordinate in the rotation-axis direction. Thus, if the viscous damping coefficient tensor is isotropic; i.e., E, = &,I, and se, is denoted by p, and D, = 0 (no internal damping), one has

Therefore, one may state the following:

A conservatively loaded elastic shaft cannot be destabilized by external damping alone if it is of linear viscous type with isotropic damping coeffi- cient tensor and the ratio between the coefficient p(z) and mass parameter ni(z) = p(z)A(z) per unit shaft length are constant everywhere. Otherwise, the shaft may be destabilized by small external damping as well as internal damping forces.

Accordingly, for a shaft with homogeneous material property and damping coefficient proportional to cross-sectional area A, which is usually the case for air damping under small amplitude vibrations, a rotating shaft cannot be destabilized by external damping of this type. Otherwise, small external damping forces in general have a destabilizing effect on a conservatively loaded rotating shaft.

, D. A Concept of "Perfect" and "Imperfect" Systems in Nonconservative Problems of Elastic Stability

Within the context of the present study, let us define a "perfect" elastic system as one with negligibly small internal damping forces, compared with external damping forces and satisfying the nondestabilizing condition (cf. Eq. 13)

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GYROSCOPIC CONSERVATIVE CONTlNUOUS SYSTEMS 207

for arbitrary w modes and an "imperfect" system as one that does not satisfy the above conditions. Then, quasi-stability of a n undamped g.c. system or nongyroscopic circulatory system' represents the limiting case of asymptotic stability for the corresponding "perfect" system, as the external damping force parameter (and also circulatory force parameter in the gyroscopic case) approaches zero.

There exists a wide class of "perfect" elastic systems in practical problems of importance. For example, an axially loaded rotating shaft under the influence of relatively large external damping forces (compared with internal damping forces) is such a system. Another example is the flutter problem of a constant thickness panel situated in a supersonic airflow in which the mass and aerodynamic damping coefficient (which is usually much larger than in- ternal damping coefficient in normal flight range, say less than 50,000 ft) are proportional everywhere, etc. Therefore, the quasi-stability analysis results, which are usually simpler to obtain than the corresponding asymptotic results, are useful in providing asymptotic stability results for a wide class of dynamic instability problems; i.e., the entire class of dynamic instability problems of "perfect" systems. However, before performing a quasi-stability analysis, care must be exercised to make sure in advance that the system is "perfect," because nonconservative problems of elastic stability are in general "imperfection-sensitive" if the system is "imperfect" as defined above.

IV. DISCRETE SYSTEMS AND EXAMPLE

The linear equations of motion for the general discrete system correspond- ing to the continuous system studied previously can be written as

where [m] is the mass matrix, [dl and [g] are the symmetric (damping) and antisymmetric (gyroscopic) parts of the velocity-dependent force matrix, respectively, [c] and I:n] are the symmetric (conservative) and antisymmetric (purely nonconservative; i.e., circulatory) parts of the displacement-depend- ent force matrix, and {d} is the displacement vector. The functionals F = K, R, or P in Eqs. 5a, 5b, and 6a are now given by

'It can be shown that R/2K = k , = const. is also a sufficient condition for the non- destabilizing effect of small damping on a nongyroscopic circulatory system.

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SHIEH

and J ( = G or N ) are

where i = J--i, { l v ) is the modal vector, {w*} is the complex conjugate of {IV), and superscript t stands for "transpose" of a vector or matrix. A suffi- cient condition for no destabilizing effect of small damping and circulatory forces, according to Theorems 1 and 2, is

[dl = 2k, [ml , [n] = k , [ g ] or o = G/2K = N / R ({w} = arbitrary)

(20)

where k , is a n arbitrary proportional constant. Equation 20 essentially states that if the damping matrix is proportional to twice the mass matrix and if the circulatory force matrix is proportional (by the same proportional con- stant) to the gyroscopic force matrix, small damping and circulatory forces do not destabilize an otherwise stable g.c. system. Otherwise, the former in general destabilize the latter.

Example: Consider a rotating mass-spring-dashpot system shown in Fig. 1, which is under the action of external axial (conservative) and damping forces. With reference to Fig. 1, the equations of motion are

Fig. 1 An externally loaded, rotating mass-dashpot-spring system.

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GYROSCOPIC CONSERVATIVE CONTINUOUS SYSTEMS 209

where v, and v2 are external viscous damping coefficients. Define

where k, p, and v are positive constants (parameters) with values ranging from 0 to I . Use of the standard eigenvalue solution method leads to

where

s = (m/k,)112i, a, = 2d(l + y), d = v,/@L1

a, = 2(1 + gZ - p) + d2[(y + 1)2 - (yp + v ) ~ ]

a3 = W ( Y + 1)(1 - P) - (Y - 1)s2 - (yp + v)kl (224 a4 = (1 - p - s2)' - k2 + (1 - v2)d2g2

Y = P,/v., P = Plk,, g = (mlkJ112Q

The Routh-Hurwitz criterion for stability implies

with a, = 0 and F = 0 representing divergence and flutter critical conditions. Thus, for the small damping case with p, and pJ denoting divergence and flutter critical loads, respectively,

For the externally damped (but noninternally damped) case (d << 1, y = O),

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210 SHIEH

For the undamped case, F, = a$ - 4a, > 0 for quasi-stability and FQ =O for the quasi-critical condition. Thus,

Equations 25 and 26 can be put in a more familiar form by first introducing

and substituting into these equations to obtain

where subscripts A and Q are used to denote the quantity corresponding to asymptotic (A-) and quasi (Q-) critical cases, respectively.

The quasi (Q-) and asymptotic (A-) stability/instability regions correspond- ing to the undamped case with k = 5 and the corresponding small external damping case ( d Z << 1) with the damping ratio T = 5.83 are plotted in Fig. 2. Here we see that the A-flutter region penetrates into both Q-stable and Q- divergence regions, but not vice versa. Thus, contrary to common belief, small external damping forces are seen to destabilize the conservatively loaded rotating system, provided that p # 0. If p = 0; i.e., the case of an unloaded rotating system, no flutter instability can occur for both the un- damped system and the externally damped system.

From Eq. 28 we also observe that small external damping forces d o not destabilize the undamped rotating system for the following two cases:

The nondestabilizing effect for the second case is predictable because, with T = 1, the nondestabilizingcondition of Eq. 20 is satisfied (with k , = 2 ~ , / r n ) . ~

'The nondestabilizing effect for the first case (k = 1) is also readily predictable, because in this case o = -j at a critical state and i t can be shown to satisfy the second condition of Eq. 20 at the critical state.

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GYROSCOPIC CONSERVATIVE CONTINUOUS SYSTEMS

Fig. 2 Destabilizing effects of small external damping forces (k = 5, 1 = 5.83, 8, =

8 2 = 0).

As a matter of fact, the corresponding system is a "perfect" system defined in Eq. 17d.

In the small damping case in which internal damping is also present, it is rather obvious from Eq. 24 that the latter has a destabilizing effect.

V. CONCLUDING REMARKS

With emphasis on small damping and the circulatory force case, dynamic instability of an elastic or Voigt-Kelvin type viscoelastic solid body subjected to both velocity- and displacement-dependent external forces has been studied. The stability/instability conditions formulated previously [I, 21 have

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been first reviewed and subsequently used in studying the effect of small damping and circulatory forces on stability of gyroscopic conservative systems. It has been shown that these small forces in general have destabilizing effects on the latter and a sufficient condition for nondestabilizing effect of these small forces has been obtained. A concept of "perfect" systems in elastic stability of nonconservative problems has also been presented for relating the quasi-stability with the Liapunov sense of stability. For such a system, the quasi-stability of the corresponding gyroscopic conservative or non- gyroscopic circulatory system represents the limiting case in which the small damping forces, also circulatory forces in the gyroscopic case, approach zero.

An example problem of a conservatively loaded rotating system was presented for demonstration purposes. It was demonstrated that for the case of unequal spring constants and a mass matrix that is not proportional to the external damping matrix, the undamped conservatively loaded rotating system is destabilized by small external damping forces alone. This result, also shown to hold for the general case, appears to be contrary to a common belief that external damping forces alone cannot destabilize an otherwise stable gyroscopic conservative system, such as the rotating system exemplified.

REFERENCES

1. R. C. Shieh and E. Masur, Some general principles of dynamic instability of solid bodies, Z. Angew. Math. Phys. 19: 927-941 (1968).

2. R. C. Shieh, Energy and variational principles for generalized (gyroscopic) conservative problems, Inr. J. Non-Linear Mech. 5 : 594-509 (1971).

3. R. C. Shieh, Variational method in the stability analysis of non-conservative problems, 2. Angew. Math. Phys. 21: 88-100 (1970).

4. R. C. Shieh, Asymptotic stability analysis of viscoelastic systems, J . Eng. Mech. Div., ASCE E m : 193-203 (1971).

5. W. Thomson and G. Tait, A Treatise on NaturalPhilosophy, Vol. I, Part I, Cambridge, 1879, p. 370.

6. H. Ziegler, Linear elastic stability, Z. Angew. Math. Phys. 4 : 89, 167 (1953). 7. H. Ziegler, On the concept of elastic stability, Adv. Appl. Mech. 4: 351 (1956). 8. J. Beth, On the stabilization of instable equilibrium by means of gyroslopic forces,

Philos. Mag. 49: 447 (1925). 9. 0. Bottema, On the stability of equilibrium of a linear mechanical system, Z. Angew.

Math. Phys. (1955). 10. K. Huseyin and R. Plaut, Transverse vibrations and stability of systems with gyro-

scopic forces, J. Struct. Mech. 3(2): 163-177 (1974). I I . K. Huseyin and R. Plaut, Divergence and flutter boundaries of systems under corn-

bined conservative and gyroscopic forces, IUTAM Symposium on Dynamics of Rotors, LynbyjDenmark, August 12-16, 1974, pp. 182-204.

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GYROSCOPIC CONSERVATIVE CONTINUOUS SYSTEMS 213

12. K. Huseyin, Vibrations and Sfabilify of Mulliple-Paramefer Sysfems, Sijthoff & Noordhoff, 1978.

13. R. C. Shieh, Some principles of elastic shaft stability including variational principles, J. Appl. Mech. 104: 191-196 (1982).

14. N. Minorski, Nonlinear Oscillations, Van Nostrand, Princeton, New Jersey, 1962, pp. 144-145.

Received December 1982

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