- Email: [email protected]
The effect of dissipative and gyroscopic forces on the stability of a class of linear time-varying systems
Journal of Applied Mathematics and Mechanics 77 (2013) 278–286
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
The effect of dissipative and gyroscopic forces on the stability of a class of linear time-varying systems夽 V.I. Kalenova, V.M. Morozov Moscow, Russia
a r t i c l e
i n f o
Article history: Received 24 April 2012
a b s t r a c t It is shown that the effect of dissipative and gyroscopic forces on a certain class of potential linear timevarying system differs considerably from the effect of these forces on a time-invariant system. Examples are considered. In the first of these, the equations of motion of a disk, attached to a rotating weightless elastic shaft, are investigated, taking external friction into account. The results obtained are compared with the results obtained previously by others when considering this problem. In the second example, certain problems of the stability of rotation of a Lagrange top on a base subjected to vertical harmonic vibrations are investigated. © 2013 Elsevier Ltd. All rights reserved.
Unlike linear time-invariant mechanical systems, for which the effect of forces of different physical kinds on the stability of motion has been investigated in detail (see, for example, Refs 1-7), in the case of time-varying systems the problems related to the effect of gyroscopic and dissipative forces have hardly been considered. We only know of the well-known conclusion that it is possible to extend the region of instability when damping is introduced in the case of combination resonance in a multidimensional Hamiltonian system with periodic coefficients.8 1. The reducibility of linear time-varying mechanical systems To investigate the effect of gyroscopic and dissipative forces on the stability of time-varying systems, we will consider the case when a second-order multidimensional homogeneous time-varying system is reduced to a system with constant coefficients using a constructive transformation (see Refs 9-11, where a number of mechanical problems, the mathematical models of which belong to the class considered below, are investigated). Consider the second-order time-varying system (1.1) where x is an n × 1 state vector and Ni (t) (i = 1, 2, 3) are n × n matrices with continuously differentiable elements in the interval I, which belongs to the class of systems, the matrices of which satisfy the equation (1.2) The solutions of matrix equations (1.2) for the matrices Ni (t) have the form (1.3) In particular, if some of the matrices Nj (t) = Nj0 = const, it is necessary that Nj0 C = CNj0 . In the majority of mechanics problems the matrix N1 (t) is symmetrical. It can be shown that the matrix N1 (t), represented in the form (1.3), satisfies the symmetry properties if the matrix C is skew symmetric (CT = −C), which we will also assume.
夽 Prikl. Mat. Mekh., Vol. 77, No. 3, pp. 386–397, 2013. E-mail address: [email protected] (V.I. Kalenova). 0021-8928/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jappmathmech.2013.09.003
V.I. Kalenova, V.M. Morozov / Journal of Applied Mathematics and Mechanics 77 (2013) 278–286
279
Applying the transformation (1.4) to system (1.1), we obtain the time-invariant system9–11 (1.5) Here (1.6) Suppose, for simplicity, N1 (t) = En . In accordance with the traditional representation of a mechanical system, on which gyroscopic, dissipative, potential and non-conservative position forces act, we write the initial system (1.1) in the form (1.7) The matrices D(t) and K(t) are symmetric, G(t) and F(t) are skew symmetric, and they all, as before, satisfy Eq. (1.2). Then
We will consider the structure of time-invariant system (1.5) in more detail. For this purpose we will represent it in the form
(1.8) Note the following properties of time-independent system (1.8): it does not contain non-conservative position forces (F1 = 0), if F0 = −D0+ − G0− , and only contains potential and dissipative forces if G0 = −2C, F0 = −D0+ ; then
In particular, the time-invariant system contains only potential forces if
The following assertions hold. 1. The system containing time-varying potential forces with a matrix which satisfies Eq. (1.2), and time-invariant gyroscopic forces (D0 = 0, F0 = 0 and G(t) = G0 ), reduces to time-invariant system (1.8) of the same structure if and only if (1.9) 2. A system containing only time-varying potential forces of this form reduces to time-invariant system (1.8), containing, in addition to potential forces, also gyroscopic forces, when (1.10) 3. When a system containing time-varying potential forces of this form and constant dissipative forces with a matrix D = dE, d > 0, is reduced to time-independent system (1.8), it is necessary that, in addition to potential, dissipative and gyroscopic forces, there should also be non-conservative position forces, where
2. The stability of linear time-varying mechanical systems Transformation (1.4) is a Lyapunov transformation, since the matrices exp(Ct) and exp(−Ct) are bounded. Hence, an investigation of the stability of time-varying system (1.1) ((1.7)) reduces to an analysis of the stability of time-invariant system (1.5) (or (1.8)). The nature of the stability of system (1.1) is determined by the properties of the roots of the characteristic equation of system (1.5)
In a number of cases, the stability of system (1.1) can also be investigated by applying the Kelvin–Chetayev theorems and their generalizations1-7,12-14 to time-invariant system (1.8). Suppose there are no non-conservative position forces (F(t)≡0) in the time-varying system and the matrix of the dissipative forces has the form D(t) = dE, d = const > 0. Then system (1.7) can be written as follows: (2.1)
280
V.I. Kalenova, V.M. Morozov / Journal of Applied Mathematics and Mechanics 77 (2013) 278–286
The time-invariant system corresponding to it has the form
(2.2) As already stated in Section 1, in this case system (2.2), unlike system (2.1), contains non-conservative position forces. The case when there are no dissipative forces (d = 0). Theorem 1. Time-varying potential system (2.1) (d = 0, G0 = 0) with matrix T(t), satisfying Eq. (1.2), where CT = −C, is stable if the matrix K0 + C2 is positive definite. The proof follows from Assertion 2 and the Kelvin–Chetayev theorem. The number of negative eigenvalues of the matrix K0 + C2 is called the degree of instability of the corresponding time-invariant system (2.2) (d = 0, G0 = 0). If the degree of instability of this system is odd, systems (2.2) and (2.1) are unstable. If the degree of instability is even, gyroscopic stabilization in system (2.1) is possible, and if it is realized, time-varying system (2.1) is also stable. System (2.2) when d = 0 and condition (1.9) is satisfied has the energy integral
to which the following time-varying integral of initial time-varying system (2.1) corresponds
The following theorem holds. Theorem 2. Time-varying system (2.1) when d = 0 and condition (1.9) is satisfied is stable if the matrix K1 = K0 + C2 + G0 C is positive definite. If the system is time-invariant, then, by the Kelvin–Chetayev theorem, the introduction of arbitrary gyroscopic forces into the stable / 0 into a time-varying potential system does not disturb its stability. The introduction of gyroscopic forces with the constant matrix G0 = potential system, may considerably change the stability property of the initial time-varying system, since the presence of the matrix G0 may change the properties of the matrix K1 . Theorem 3. If d = 0, detC = / 0, detG0 = / 0 in system (2.1) and condition (1.9) is satisfied, this system can always be made stable by an appropriate choice of the matrix G0 . Proof. The orthogonal transformation y = Tz, by virtue of condition (1.9), simultaneously converts matrices G0 and C of system (2.2) to partitioned-diagonal form15
n = 2m, ± is (s > 0) are eigenvalues of the matrix C, and gs > 0(s = 1, . . ., m) Hence
Hence it follows that, for sufficiently large values of gs , the matrix TT K1 T will always be positive definite irrespective of the properties of the matrices K0 and C. Then, in time-invariant system (2.2), the matrix of the potential forces TT K1 T is positive definite, the matrix TT G1 T is skew-symmetric, and by the Kelvin–Chetayev theorem, system (2.2), and consequently also system (2.1), are stable. On the other hand, as examples show (see below), by an appropriate choice of the matrix G0 system (2.2) ((2.1)) can be made unstable, i.e., gyroscopic destabilization is possible. The effect of dissipative forces (d = / 0). Theorem 4. Time-varying system (2.1) (provided that the matrices G(t) and K(t) satisfy Eq. (1.2)) and the reduced time-invariant system (2.2) corresponding to it, are asymptotically stable if the matrix
is positive definite. Proof.
We construct the Lyapunov function in the form
The condition for the function V to be positive definite is15 the condition for the matrix 2 − T1 1 > 0, i.e., the matrix
V.I. Kalenova, V.M. Morozov / Journal of Applied Mathematics and Mechanics 77 (2013) 278–286
281
to be positive definite. The derivative of the function V, calculated by virtue of Eqs (2.2), can be represented in the form
The function V˙ is negative definite if the matrix R1 is positive definite. With this condition, the function V is positive definite. Hence, the function V, when the conditions of Theorem 4 are satisfied, satisfies the conditions of Lyapunov’s theorem on asymptotic stability. Corollary. If condition (1.9) is satisfied, time-varying system (2.1) and the time-invariant system (2.2) corresponding to it are asymptot˜ 1 is positive definite. ically stable provided that the matrix R A time-invariant system with the following special structure was considered in Ref. 5 (2.3) System (2.3) corresponds to time-invariant system (2.2) when
The conditions for system (2.3) to be asymptotically stable have the form5
It can be shown, bearing in mind the equalities
that these conditions follow from the conditions for time-invariant system (2.2) to be stable, formulated in the corollary of Theorem 4. Theorem 5.
If the matrix
is positive definite, time-invariant system (2.2) and time-varying system (2.1) are unstable (provided the matrices G(t) and K(t) satisfy Eq. (1.2)). Proof.
Consider the function
Its derivative, by virtue of system (2.2), can be represented in the form
The function W will be positive definite if the matrix R2 is positive definite. Hence, the function W, with the conditions of Theorem 5, satisfies Lyapunov’s theorem on stability. The special case when there are no gyroscopic forces (G(t) = 0). Suppose the following conditions are satisfied16 in time-varying system (2.1) (2.4) Then, by taking the specific features of the structure of the matrices of reduced time-invariant system (2.2) into account, we can formulate the following theorem. Theorem 6. Suppose that, in time-varying system (2.1), G(t) = 0, while the matrix K(t) satisfies Eq. (1.2). Then, if conditions (2.4) are / 0, system (2.1) and the corresponding time-invariant system (2.2) are asymptotically stable if and only if the following satisfied and detC = conditions are satisfied
(2.5) The proof of the theorem is based on the following lemma. Lemma (16 ). If conditions (2.4) are satisfied, an orthogonal matrix T exists such that the symmetric matrix K1 and the skew-symmetric matrix C (detC = / 0) may be reduced simultaneously to the partitioned-diagonal form:
282
V.I. Kalenova, V.M. Morozov / Journal of Applied Mathematics and Mechanics 77 (2013) 278–286
were (n = 2m, ±is (s >0)) are the eigenvalues of the matrix C. Using the transformation y = Tz, system (2.2) is reduced to a set of m independent second-order subsystems
(2.6) Conditions (2.5) of Theorem 6 for system (2.6) have the form
(2.7) On the other hand, the use of the Gurvits criterion for the characteristic equation of each subsystem (2.6) gives the same conditions (2.7) as the necessary and sufficient conditions for asymptotic stability. Remark. Time-invariant system (2.2), which is acted upon by dissipative, gyroscopic and non-conservative position forces (K1 = 0, K0 = −C2 ) and which has the form
is asymptotically stable, since the conditions of Theorem 6 are satisfied. When there are no dissipative forces (d = 0) we have the following theorem. / 0 and conditions (2.4) are Theorem 7. Time-varying potential system (2.1) with matrix K(t), which satisfies Eq. (1.2), for which detC = satisfied, is stable if (2.8) Conditions (2.8) are the stability conditions of time-invariant system (2.2) and correspond to known conditions of stability.16 Theorems 4 – 6 remain true like the theorems on the stability and instability in the first approximation of the zero solution of the corresponding non-linear systems. Theorems 1 – 7 can in fact be considered as theorems on the stability of time-invariant systems of special structure. 3. Examples Example 1.
Consider the equations
(3.1) They can be treated as the equations of motion, written in a fixed system of coordinates, of a disk fastened to a weightless elastic shaft, rotating with constant angular velocity , taking external friction into account, which is characterized by a matrix of the dissipated force N2 with coefficient d ≥ 0. An axial compressive force also acts on the shaft, which has unequal stiffnesses c1 and c2 in two mutually perpendicular directions. The matrix of the position forces N3 (t) satisfies Eq. (1.2) when C = J. Replacement of variables (1.3), which represents a transition to a rotating system of coordinates, reduces system (3.1) to time-invariant form
(3.2) Hence, as indicated above, when changing to a time-invariant system the presence of dissipative forces in initial time-varying system (3.1) means that, in time-invariant system (3.2), non-conservative position forces also occur in addition to dissipative forces. When there is no dissipation in system (3.1), position forces are replaced by potential forces. Equations (3.2) are identical with the equations of motion of this shaft, considered earlier in Ref. 17, when there are no compressive forces ( = 0), written in the mobile system of coordinates, with e = d, i = 0, 2j = ␣ ± , where e and i are the coefficients of
external and internal friction and j are the partial frequencies of the non-rotating shaft, calculated ignoring friction. When there are no friction forces (d = 0) these same Eqs (3.2) are identical, apart from the notation, with the known equations considered in a number of papers (see, for example, Refs 18 and 19), where the boundaries of the regions of stability and instability were investigated as a function of the values of the parameters c1 , c2 , and .
V.I. Kalenova, V.M. Morozov / Journal of Applied Mathematics and Mechanics 77 (2013) 278–286
283
Fig. 1.
Digressing from the mechanical analogies given above, we will consider the regions of stability in this example as a function of the parameters , ␣,  and d of initial system (3.1), integrating the parameters and  as the frequency and amplitude of the parametric action. The application of Theorem 4 to this problem leads to the following sufficient conditions for stability (3.3) According to Theorem 6 the necessary and sufficient conditions for the asymptotic stability of system (3.1) have the form (3.4) (These conditions are obtained directly from the Gurvits criterion as it applies to time-invariant system (3.2).) Obviously, conditions (3.3) are much stricter than conditions (3.4). When  = 0, system (3.1) becomes time-invariant. When ␣ > 0 and d = 0 we have stability, which, when d > 0, becomes asymptotic. When ␣ < 0 and d = 0 the system is unstable and the degree of instability is even. The case when there are no dissipative forces (d = 0). The matrix K1 is positive definite when the following condition is satisfied (we assume that  > 0) (3.5) which can only occur when ␣ > 0. This is the condition of secular stability. If  > |␣ − 2 |, the degree of instability of system (3.2) is odd, and system (3.2) and, consequently, system (3.1) are unstable irrespective of the presence of gyroscopic forces. If  < |␣ − 2 |, the degree of instability of system (3.2) is even, and gyroscopic stabilisation is possible, which is realized, as follows from the necessary and sufficient conditions for stability, which have the form (3.6) Thus, in the initial time-varying system with potential periodic systems of a certain structure, gyroscopic stabilization is possible for certain values of the frequency and amplitude  of the periodic forces. Note that, in this system with periodic coefficients when ␣ > 0 in the ,  parameter plane, only one region of instability (D + C) exists (see Fig. 1) unlike, for example, the well-known Mathieu–Hill equation, for which the number of instability zones is infinite. The effect of dissipative forces (d > 0). When ␣ > 0 it follows from inequalities (3.4) that the region of instability is extended compared with the case when there are no dissipative forces and a minimum value of the excitation amplitude * exists, for which an instability region D occurs, i.e., in this case the presence of dissipative forces cannot lead to instability. In Fig. 1 we show, in the ,  plane, the regions of stability for ␣ = 0.5 and values of the friction coefficient d = 1.2 and d = 0.4. When there are no dissipative forces, A and B are regions of stability, where A is a region of secular stability, B is a region of gyroscopic stabilization, and C and D are regions of instability. When there are dissipative forces, A, B and C are regions of asymptotic stability and D is a region of instability. When ␣ < 0 the regions of stability in the ,  parameter plane, defined by inequalities (3.4) and (3.6), are shown in Fig. 2 for d = 0 (regions A and C) and for d > 0 (regions A and B). It follows from a comparison of these regions that a region C exists where the presence of dissipative forces leads to instability for values of the parameters and  for which stability occurred when there was no dissipation. Region C is defined by the inequalities
In region B, the presence of dissipative forces ensures asymptotic stability where instability occurred when d = 0.
284
V.I. Kalenova, V.M. Morozov / Journal of Applied Mathematics and Mechanics 77 (2013) 278–286
Fig. 2.
The effect of gyroscopic forces. To consider the effect of gyroscopic forces on the behaviour of the solutions of the time-varying potential system in this example we will introduce in Eq. (3.1) gyroscopic forces with a matrix
assuming that there are no dissipative forces (d = 0). In this case, time-invariant system (3.2) takes the form
(3.7) The sufficient conditions for secular stability are the conditions for the matrix K1 to be positive definite: (3.8) from which, according to Theorem 3, it follows that they are always satisfied for a sufficiently large parameter g > 0. This indicates that, by adding sufficiently large gyroscopic forces to a time-varying potential system of this structure, one can make such a system stable, in the secular sense. On the other hand, the parameter g can always be chosen so that inequalities (3.8) have different signs. In this case system (3.7) would have an odd degree of instability and, consequently, would be unstable, irrespective of all the remaining parameters of the problem, i.e., irrespective of whether the initial system is stable or unstable when there are no additional gyroscopic forces. Hence, both gyroscopic stabilization and destabilization are possible in the system. The necessary and sufficient conditions for stability, obtained both on the basis of Theorem 7, and also from an analysis of the characteristic equation, have the form (3.9) Suppose ␣ ≥ 1. From conditions (9) we have the unique stability condition (3.10) The regions of stability for g = 0 and g = 1 are shown in Fig. 3. When g ≡ 0( < |␣ − 2 |) the system is stable in regions A, B, C and E of values of the parameters of and . When g = 1 the system is stable in regions A, B, C and D. In the case when g > 0, for any value of the parameter , there is always a region of values of , of which gyroscopic destabilisation occurs (region E). In region D there is gyroscopic stabilization for limited values of the excitation amplitude ( <  *) and in a certain range of variation of the frequency . Stability is preserved in regions A, B and C. We can similarly consider the case when ␣ < 0. We will consider one other example, which demonstrates the particular features of the effect of gyroscopic forces on a time-varying system. Example 2.
The system
V.I. Kalenova, V.M. Morozov / Journal of Applied Mathematics and Mechanics 77 (2013) 278–286
285
Fig. 3.
(3.11) describes, in the first approximation, the perturbed motion of a Lagrange gyroscope, set up on a base, performing a vertical harmonic vibration.20 When g = 0 it is split into two well-known Mathieu equations (see, for example, Refs 21 and 22) (3.12) for which a countless set of alternating regions of stability and instability exist in the ␦, parameter plane. Suppose g = / 0. Then, the replacement of variables x = exp(−Gt)y reduces system (3.11) to the form (3.13) The stability of system (3.13) is equivalent to the stability of system (3.11), since the conversion matrix of exp(−Gt) is bounded (GT = −G). One of the regions of stability for system (3.12) is defined by the inequalities22
(3.14) Suppose the parameters ␦ and belong to stability region (3.14). The parameter g can then be chosen so that the following inequality is satisfied
which defines the region of instability for system (3.13). This implies gyroscopic destabilization. By increasing the value of the parameter g, we can enter the second region of stability, defined by the inequalities
Further, by increasing the parameter g we can alternately enter regions of instability and stability. Conversely, if the parameters ␦ and belong to some regions of instability, we can always choose a value of the parameter g so that gyroscopic stabilization occurs. These examples demonstrate the important difference between the properties of time-varying systems and time-invariant systems: the effect of dissipative and gyroscopic forces on a time-varying potential system differs considerably from the effect of the same forces on a time-invariant potential system. Acknowledgements This research was supported by the Russian Foundation for Basic Research (12-01-00371).
286
V.I. Kalenova, V.M. Morozov / Journal of Applied Mathematics and Mechanics 77 (2013) 278–286
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.
Thomson W, Tait P. Treatise on Natural Philosophy L. Cambridge: Univ. Press; 1879. Pt 1, 285 p. Chetayev NG. The Stability of Motion. Papers on Analytical Mechanics. Moscow: Izd Akad Nauk SSSR; 1962. Merkin DR. Gyroscopic Systems. Moscow: Nauka; 1974. Karapetyan AV, Rumyantsev VV. The stability of conservative and dissipative systems. Advances in Science and Technology. General Mechanics, vol. 6. Moscow: VINITI; 1983. Agafonov SA. Stability and motion stabilization of nonconservative mechanical systems. J Math Sci Dynamical systems II 2002;112(5):4419–97. Walker JA. Stability of linear conservative gyroscopic stystems. Trans ASME J Appl Mech 1991;58(1):229–32. Bulatoviˇc RM. The stability of linear potential gyroscopic systems when the potential energy has a maximum. J Appl Math Mech 1997;61(3):371–6. Yakubovich VA. Starzhinskii. Parametric Resonance in Linear Systems. Moscow: Nauka; 1987. Kalenova VI, Morozov VM. Linear Time-Varying Systems and Their Applications to Mechanics Problems. Moscow: Fizmatlit; 2010. Kalenova VI, Morozov VM, Sobolevskii PM. The stability of a specific class of mechanical systems. J Appl Math Mech 2008;72(2):152–8. Kalenova VI, Morozov VM. The reducibility of second-order linear time-varying homogeneous systems. J Appl Math Mech 2011;75(6):647–51. Bulatovic RM. A note on the damped vibrating systems. J Theoret Appl Belgrade 2006;33(3):213–21. Krechetnikov R, Marsden JE. Dissipation-induced instabilities in finite dimensions. Rev Modern Phys 2007;79:519–53. Kirillov ON. Gyroscopic Stabilization in presence of non-conservative forces. In: 12th IFT. MM World Congress Besancon. 2007. p. 780–5. Horn RA, Johnson ChR. Matrix Analysis. Cambridge: Univ. Press; 1985. Bulatovic RM. A stability theorem for gyroscopic systems. Acta Mechanica Springer 1999;136(1–2):119–23. Bolotin VV. Non-Conservative Problems of Elastic Stability Theory. Moscow: Nauka; 1961. Huseyin K, Hagedorn P, Teschner W. On the stability of linear conservative gyroscopic systems. ZAMP 1983;34(6):807–15. Seyranian AP, Stoustrup J, Kliem W. On gyroscopic stabilization. ZAMP 1995;46:255–67. Koshlyakov VN. Structural transformations of dynamical systems with gyroscopic forces. J Appl Math Mech 1997;61(5):751–6. Merkin DR. Introduction to the Theory of Stability of Motion. Moscow: Nauka; 1976. Stoker JJ. Nonlinear Vibration in Mechanical and Electrical Systems. N.Y.: L. Interscience; 1950.
Translated by R.C.G.
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
The effect of dissipative and gyroscopic forces on the stability of a class of linear time-varying systems夽 V.I. Kalenova, V.M. Morozov Moscow, Russia
a r t i c l e
i n f o
Article history: Received 24 April 2012
a b s t r a c t It is shown that the effect of dissipative and gyroscopic forces on a certain class of potential linear timevarying system differs considerably from the effect of these forces on a time-invariant system. Examples are considered. In the first of these, the equations of motion of a disk, attached to a rotating weightless elastic shaft, are investigated, taking external friction into account. The results obtained are compared with the results obtained previously by others when considering this problem. In the second example, certain problems of the stability of rotation of a Lagrange top on a base subjected to vertical harmonic vibrations are investigated. © 2013 Elsevier Ltd. All rights reserved.
Unlike linear time-invariant mechanical systems, for which the effect of forces of different physical kinds on the stability of motion has been investigated in detail (see, for example, Refs 1-7), in the case of time-varying systems the problems related to the effect of gyroscopic and dissipative forces have hardly been considered. We only know of the well-known conclusion that it is possible to extend the region of instability when damping is introduced in the case of combination resonance in a multidimensional Hamiltonian system with periodic coefficients.8 1. The reducibility of linear time-varying mechanical systems To investigate the effect of gyroscopic and dissipative forces on the stability of time-varying systems, we will consider the case when a second-order multidimensional homogeneous time-varying system is reduced to a system with constant coefficients using a constructive transformation (see Refs 9-11, where a number of mechanical problems, the mathematical models of which belong to the class considered below, are investigated). Consider the second-order time-varying system (1.1) where x is an n × 1 state vector and Ni (t) (i = 1, 2, 3) are n × n matrices with continuously differentiable elements in the interval I, which belongs to the class of systems, the matrices of which satisfy the equation (1.2) The solutions of matrix equations (1.2) for the matrices Ni (t) have the form (1.3) In particular, if some of the matrices Nj (t) = Nj0 = const, it is necessary that Nj0 C = CNj0 . In the majority of mechanics problems the matrix N1 (t) is symmetrical. It can be shown that the matrix N1 (t), represented in the form (1.3), satisfies the symmetry properties if the matrix C is skew symmetric (CT = −C), which we will also assume.
夽 Prikl. Mat. Mekh., Vol. 77, No. 3, pp. 386–397, 2013. E-mail address: [email protected] (V.I. Kalenova). 0021-8928/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jappmathmech.2013.09.003
V.I. Kalenova, V.M. Morozov / Journal of Applied Mathematics and Mechanics 77 (2013) 278–286
279
Applying the transformation (1.4) to system (1.1), we obtain the time-invariant system9–11 (1.5) Here (1.6) Suppose, for simplicity, N1 (t) = En . In accordance with the traditional representation of a mechanical system, on which gyroscopic, dissipative, potential and non-conservative position forces act, we write the initial system (1.1) in the form (1.7) The matrices D(t) and K(t) are symmetric, G(t) and F(t) are skew symmetric, and they all, as before, satisfy Eq. (1.2). Then
We will consider the structure of time-invariant system (1.5) in more detail. For this purpose we will represent it in the form
(1.8) Note the following properties of time-independent system (1.8): it does not contain non-conservative position forces (F1 = 0), if F0 = −D0+ − G0− , and only contains potential and dissipative forces if G0 = −2C, F0 = −D0+ ; then
In particular, the time-invariant system contains only potential forces if
The following assertions hold. 1. The system containing time-varying potential forces with a matrix which satisfies Eq. (1.2), and time-invariant gyroscopic forces (D0 = 0, F0 = 0 and G(t) = G0 ), reduces to time-invariant system (1.8) of the same structure if and only if (1.9) 2. A system containing only time-varying potential forces of this form reduces to time-invariant system (1.8), containing, in addition to potential forces, also gyroscopic forces, when (1.10) 3. When a system containing time-varying potential forces of this form and constant dissipative forces with a matrix D = dE, d > 0, is reduced to time-independent system (1.8), it is necessary that, in addition to potential, dissipative and gyroscopic forces, there should also be non-conservative position forces, where
2. The stability of linear time-varying mechanical systems Transformation (1.4) is a Lyapunov transformation, since the matrices exp(Ct) and exp(−Ct) are bounded. Hence, an investigation of the stability of time-varying system (1.1) ((1.7)) reduces to an analysis of the stability of time-invariant system (1.5) (or (1.8)). The nature of the stability of system (1.1) is determined by the properties of the roots of the characteristic equation of system (1.5)
In a number of cases, the stability of system (1.1) can also be investigated by applying the Kelvin–Chetayev theorems and their generalizations1-7,12-14 to time-invariant system (1.8). Suppose there are no non-conservative position forces (F(t)≡0) in the time-varying system and the matrix of the dissipative forces has the form D(t) = dE, d = const > 0. Then system (1.7) can be written as follows: (2.1)
280
V.I. Kalenova, V.M. Morozov / Journal of Applied Mathematics and Mechanics 77 (2013) 278–286
The time-invariant system corresponding to it has the form
(2.2) As already stated in Section 1, in this case system (2.2), unlike system (2.1), contains non-conservative position forces. The case when there are no dissipative forces (d = 0). Theorem 1. Time-varying potential system (2.1) (d = 0, G0 = 0) with matrix T(t), satisfying Eq. (1.2), where CT = −C, is stable if the matrix K0 + C2 is positive definite. The proof follows from Assertion 2 and the Kelvin–Chetayev theorem. The number of negative eigenvalues of the matrix K0 + C2 is called the degree of instability of the corresponding time-invariant system (2.2) (d = 0, G0 = 0). If the degree of instability of this system is odd, systems (2.2) and (2.1) are unstable. If the degree of instability is even, gyroscopic stabilization in system (2.1) is possible, and if it is realized, time-varying system (2.1) is also stable. System (2.2) when d = 0 and condition (1.9) is satisfied has the energy integral
to which the following time-varying integral of initial time-varying system (2.1) corresponds
The following theorem holds. Theorem 2. Time-varying system (2.1) when d = 0 and condition (1.9) is satisfied is stable if the matrix K1 = K0 + C2 + G0 C is positive definite. If the system is time-invariant, then, by the Kelvin–Chetayev theorem, the introduction of arbitrary gyroscopic forces into the stable / 0 into a time-varying potential system does not disturb its stability. The introduction of gyroscopic forces with the constant matrix G0 = potential system, may considerably change the stability property of the initial time-varying system, since the presence of the matrix G0 may change the properties of the matrix K1 . Theorem 3. If d = 0, detC = / 0, detG0 = / 0 in system (2.1) and condition (1.9) is satisfied, this system can always be made stable by an appropriate choice of the matrix G0 . Proof. The orthogonal transformation y = Tz, by virtue of condition (1.9), simultaneously converts matrices G0 and C of system (2.2) to partitioned-diagonal form15
n = 2m, ± is (s > 0) are eigenvalues of the matrix C, and gs > 0(s = 1, . . ., m) Hence
Hence it follows that, for sufficiently large values of gs , the matrix TT K1 T will always be positive definite irrespective of the properties of the matrices K0 and C. Then, in time-invariant system (2.2), the matrix of the potential forces TT K1 T is positive definite, the matrix TT G1 T is skew-symmetric, and by the Kelvin–Chetayev theorem, system (2.2), and consequently also system (2.1), are stable. On the other hand, as examples show (see below), by an appropriate choice of the matrix G0 system (2.2) ((2.1)) can be made unstable, i.e., gyroscopic destabilization is possible. The effect of dissipative forces (d = / 0). Theorem 4. Time-varying system (2.1) (provided that the matrices G(t) and K(t) satisfy Eq. (1.2)) and the reduced time-invariant system (2.2) corresponding to it, are asymptotically stable if the matrix
is positive definite. Proof.
We construct the Lyapunov function in the form
The condition for the function V to be positive definite is15 the condition for the matrix 2 − T1 1 > 0, i.e., the matrix
V.I. Kalenova, V.M. Morozov / Journal of Applied Mathematics and Mechanics 77 (2013) 278–286
281
to be positive definite. The derivative of the function V, calculated by virtue of Eqs (2.2), can be represented in the form
The function V˙ is negative definite if the matrix R1 is positive definite. With this condition, the function V is positive definite. Hence, the function V, when the conditions of Theorem 4 are satisfied, satisfies the conditions of Lyapunov’s theorem on asymptotic stability. Corollary. If condition (1.9) is satisfied, time-varying system (2.1) and the time-invariant system (2.2) corresponding to it are asymptot˜ 1 is positive definite. ically stable provided that the matrix R A time-invariant system with the following special structure was considered in Ref. 5 (2.3) System (2.3) corresponds to time-invariant system (2.2) when
The conditions for system (2.3) to be asymptotically stable have the form5
It can be shown, bearing in mind the equalities
that these conditions follow from the conditions for time-invariant system (2.2) to be stable, formulated in the corollary of Theorem 4. Theorem 5.
If the matrix
is positive definite, time-invariant system (2.2) and time-varying system (2.1) are unstable (provided the matrices G(t) and K(t) satisfy Eq. (1.2)). Proof.
Consider the function
Its derivative, by virtue of system (2.2), can be represented in the form
The function W will be positive definite if the matrix R2 is positive definite. Hence, the function W, with the conditions of Theorem 5, satisfies Lyapunov’s theorem on stability. The special case when there are no gyroscopic forces (G(t) = 0). Suppose the following conditions are satisfied16 in time-varying system (2.1) (2.4) Then, by taking the specific features of the structure of the matrices of reduced time-invariant system (2.2) into account, we can formulate the following theorem. Theorem 6. Suppose that, in time-varying system (2.1), G(t) = 0, while the matrix K(t) satisfies Eq. (1.2). Then, if conditions (2.4) are / 0, system (2.1) and the corresponding time-invariant system (2.2) are asymptotically stable if and only if the following satisfied and detC = conditions are satisfied
(2.5) The proof of the theorem is based on the following lemma. Lemma (16 ). If conditions (2.4) are satisfied, an orthogonal matrix T exists such that the symmetric matrix K1 and the skew-symmetric matrix C (detC = / 0) may be reduced simultaneously to the partitioned-diagonal form:
282
V.I. Kalenova, V.M. Morozov / Journal of Applied Mathematics and Mechanics 77 (2013) 278–286
were (n = 2m, ±is (s >0)) are the eigenvalues of the matrix C. Using the transformation y = Tz, system (2.2) is reduced to a set of m independent second-order subsystems
(2.6) Conditions (2.5) of Theorem 6 for system (2.6) have the form
(2.7) On the other hand, the use of the Gurvits criterion for the characteristic equation of each subsystem (2.6) gives the same conditions (2.7) as the necessary and sufficient conditions for asymptotic stability. Remark. Time-invariant system (2.2), which is acted upon by dissipative, gyroscopic and non-conservative position forces (K1 = 0, K0 = −C2 ) and which has the form
is asymptotically stable, since the conditions of Theorem 6 are satisfied. When there are no dissipative forces (d = 0) we have the following theorem. / 0 and conditions (2.4) are Theorem 7. Time-varying potential system (2.1) with matrix K(t), which satisfies Eq. (1.2), for which detC = satisfied, is stable if (2.8) Conditions (2.8) are the stability conditions of time-invariant system (2.2) and correspond to known conditions of stability.16 Theorems 4 – 6 remain true like the theorems on the stability and instability in the first approximation of the zero solution of the corresponding non-linear systems. Theorems 1 – 7 can in fact be considered as theorems on the stability of time-invariant systems of special structure. 3. Examples Example 1.
Consider the equations
(3.1) They can be treated as the equations of motion, written in a fixed system of coordinates, of a disk fastened to a weightless elastic shaft, rotating with constant angular velocity , taking external friction into account, which is characterized by a matrix of the dissipated force N2 with coefficient d ≥ 0. An axial compressive force also acts on the shaft, which has unequal stiffnesses c1 and c2 in two mutually perpendicular directions. The matrix of the position forces N3 (t) satisfies Eq. (1.2) when C = J. Replacement of variables (1.3), which represents a transition to a rotating system of coordinates, reduces system (3.1) to time-invariant form
(3.2) Hence, as indicated above, when changing to a time-invariant system the presence of dissipative forces in initial time-varying system (3.1) means that, in time-invariant system (3.2), non-conservative position forces also occur in addition to dissipative forces. When there is no dissipation in system (3.1), position forces are replaced by potential forces. Equations (3.2) are identical with the equations of motion of this shaft, considered earlier in Ref. 17, when there are no compressive forces ( = 0), written in the mobile system of coordinates, with e = d, i = 0, 2j = ␣ ± , where e and i are the coefficients of
external and internal friction and j are the partial frequencies of the non-rotating shaft, calculated ignoring friction. When there are no friction forces (d = 0) these same Eqs (3.2) are identical, apart from the notation, with the known equations considered in a number of papers (see, for example, Refs 18 and 19), where the boundaries of the regions of stability and instability were investigated as a function of the values of the parameters c1 , c2 , and .
V.I. Kalenova, V.M. Morozov / Journal of Applied Mathematics and Mechanics 77 (2013) 278–286
283
Fig. 1.
Digressing from the mechanical analogies given above, we will consider the regions of stability in this example as a function of the parameters , ␣,  and d of initial system (3.1), integrating the parameters and  as the frequency and amplitude of the parametric action. The application of Theorem 4 to this problem leads to the following sufficient conditions for stability (3.3) According to Theorem 6 the necessary and sufficient conditions for the asymptotic stability of system (3.1) have the form (3.4) (These conditions are obtained directly from the Gurvits criterion as it applies to time-invariant system (3.2).) Obviously, conditions (3.3) are much stricter than conditions (3.4). When  = 0, system (3.1) becomes time-invariant. When ␣ > 0 and d = 0 we have stability, which, when d > 0, becomes asymptotic. When ␣ < 0 and d = 0 the system is unstable and the degree of instability is even. The case when there are no dissipative forces (d = 0). The matrix K1 is positive definite when the following condition is satisfied (we assume that  > 0) (3.5) which can only occur when ␣ > 0. This is the condition of secular stability. If  > |␣ − 2 |, the degree of instability of system (3.2) is odd, and system (3.2) and, consequently, system (3.1) are unstable irrespective of the presence of gyroscopic forces. If  < |␣ − 2 |, the degree of instability of system (3.2) is even, and gyroscopic stabilisation is possible, which is realized, as follows from the necessary and sufficient conditions for stability, which have the form (3.6) Thus, in the initial time-varying system with potential periodic systems of a certain structure, gyroscopic stabilization is possible for certain values of the frequency and amplitude  of the periodic forces. Note that, in this system with periodic coefficients when ␣ > 0 in the ,  parameter plane, only one region of instability (D + C) exists (see Fig. 1) unlike, for example, the well-known Mathieu–Hill equation, for which the number of instability zones is infinite. The effect of dissipative forces (d > 0). When ␣ > 0 it follows from inequalities (3.4) that the region of instability is extended compared with the case when there are no dissipative forces and a minimum value of the excitation amplitude * exists, for which an instability region D occurs, i.e., in this case the presence of dissipative forces cannot lead to instability. In Fig. 1 we show, in the ,  plane, the regions of stability for ␣ = 0.5 and values of the friction coefficient d = 1.2 and d = 0.4. When there are no dissipative forces, A and B are regions of stability, where A is a region of secular stability, B is a region of gyroscopic stabilization, and C and D are regions of instability. When there are dissipative forces, A, B and C are regions of asymptotic stability and D is a region of instability. When ␣ < 0 the regions of stability in the ,  parameter plane, defined by inequalities (3.4) and (3.6), are shown in Fig. 2 for d = 0 (regions A and C) and for d > 0 (regions A and B). It follows from a comparison of these regions that a region C exists where the presence of dissipative forces leads to instability for values of the parameters and  for which stability occurred when there was no dissipation. Region C is defined by the inequalities
In region B, the presence of dissipative forces ensures asymptotic stability where instability occurred when d = 0.
284
V.I. Kalenova, V.M. Morozov / Journal of Applied Mathematics and Mechanics 77 (2013) 278–286
Fig. 2.
The effect of gyroscopic forces. To consider the effect of gyroscopic forces on the behaviour of the solutions of the time-varying potential system in this example we will introduce in Eq. (3.1) gyroscopic forces with a matrix
assuming that there are no dissipative forces (d = 0). In this case, time-invariant system (3.2) takes the form
(3.7) The sufficient conditions for secular stability are the conditions for the matrix K1 to be positive definite: (3.8) from which, according to Theorem 3, it follows that they are always satisfied for a sufficiently large parameter g > 0. This indicates that, by adding sufficiently large gyroscopic forces to a time-varying potential system of this structure, one can make such a system stable, in the secular sense. On the other hand, the parameter g can always be chosen so that inequalities (3.8) have different signs. In this case system (3.7) would have an odd degree of instability and, consequently, would be unstable, irrespective of all the remaining parameters of the problem, i.e., irrespective of whether the initial system is stable or unstable when there are no additional gyroscopic forces. Hence, both gyroscopic stabilization and destabilization are possible in the system. The necessary and sufficient conditions for stability, obtained both on the basis of Theorem 7, and also from an analysis of the characteristic equation, have the form (3.9) Suppose ␣ ≥ 1. From conditions (9) we have the unique stability condition (3.10) The regions of stability for g = 0 and g = 1 are shown in Fig. 3. When g ≡ 0( < |␣ − 2 |) the system is stable in regions A, B, C and E of values of the parameters of and . When g = 1 the system is stable in regions A, B, C and D. In the case when g > 0, for any value of the parameter , there is always a region of values of , of which gyroscopic destabilisation occurs (region E). In region D there is gyroscopic stabilization for limited values of the excitation amplitude ( <  *) and in a certain range of variation of the frequency . Stability is preserved in regions A, B and C. We can similarly consider the case when ␣ < 0. We will consider one other example, which demonstrates the particular features of the effect of gyroscopic forces on a time-varying system. Example 2.
The system
V.I. Kalenova, V.M. Morozov / Journal of Applied Mathematics and Mechanics 77 (2013) 278–286
285
Fig. 3.
(3.11) describes, in the first approximation, the perturbed motion of a Lagrange gyroscope, set up on a base, performing a vertical harmonic vibration.20 When g = 0 it is split into two well-known Mathieu equations (see, for example, Refs 21 and 22) (3.12) for which a countless set of alternating regions of stability and instability exist in the ␦, parameter plane. Suppose g = / 0. Then, the replacement of variables x = exp(−Gt)y reduces system (3.11) to the form (3.13) The stability of system (3.13) is equivalent to the stability of system (3.11), since the conversion matrix of exp(−Gt) is bounded (GT = −G). One of the regions of stability for system (3.12) is defined by the inequalities22
(3.14) Suppose the parameters ␦ and belong to stability region (3.14). The parameter g can then be chosen so that the following inequality is satisfied
which defines the region of instability for system (3.13). This implies gyroscopic destabilization. By increasing the value of the parameter g, we can enter the second region of stability, defined by the inequalities
Further, by increasing the parameter g we can alternately enter regions of instability and stability. Conversely, if the parameters ␦ and belong to some regions of instability, we can always choose a value of the parameter g so that gyroscopic stabilization occurs. These examples demonstrate the important difference between the properties of time-varying systems and time-invariant systems: the effect of dissipative and gyroscopic forces on a time-varying potential system differs considerably from the effect of the same forces on a time-invariant potential system. Acknowledgements This research was supported by the Russian Foundation for Basic Research (12-01-00371).
286
V.I. Kalenova, V.M. Morozov / Journal of Applied Mathematics and Mechanics 77 (2013) 278–286
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.
Thomson W, Tait P. Treatise on Natural Philosophy L. Cambridge: Univ. Press; 1879. Pt 1, 285 p. Chetayev NG. The Stability of Motion. Papers on Analytical Mechanics. Moscow: Izd Akad Nauk SSSR; 1962. Merkin DR. Gyroscopic Systems. Moscow: Nauka; 1974. Karapetyan AV, Rumyantsev VV. The stability of conservative and dissipative systems. Advances in Science and Technology. General Mechanics, vol. 6. Moscow: VINITI; 1983. Agafonov SA. Stability and motion stabilization of nonconservative mechanical systems. J Math Sci Dynamical systems II 2002;112(5):4419–97. Walker JA. Stability of linear conservative gyroscopic stystems. Trans ASME J Appl Mech 1991;58(1):229–32. Bulatoviˇc RM. The stability of linear potential gyroscopic systems when the potential energy has a maximum. J Appl Math Mech 1997;61(3):371–6. Yakubovich VA. Starzhinskii. Parametric Resonance in Linear Systems. Moscow: Nauka; 1987. Kalenova VI, Morozov VM. Linear Time-Varying Systems and Their Applications to Mechanics Problems. Moscow: Fizmatlit; 2010. Kalenova VI, Morozov VM, Sobolevskii PM. The stability of a specific class of mechanical systems. J Appl Math Mech 2008;72(2):152–8. Kalenova VI, Morozov VM. The reducibility of second-order linear time-varying homogeneous systems. J Appl Math Mech 2011;75(6):647–51. Bulatovic RM. A note on the damped vibrating systems. J Theoret Appl Belgrade 2006;33(3):213–21. Krechetnikov R, Marsden JE. Dissipation-induced instabilities in finite dimensions. Rev Modern Phys 2007;79:519–53. Kirillov ON. Gyroscopic Stabilization in presence of non-conservative forces. In: 12th IFT. MM World Congress Besancon. 2007. p. 780–5. Horn RA, Johnson ChR. Matrix Analysis. Cambridge: Univ. Press; 1985. Bulatovic RM. A stability theorem for gyroscopic systems. Acta Mechanica Springer 1999;136(1–2):119–23. Bolotin VV. Non-Conservative Problems of Elastic Stability Theory. Moscow: Nauka; 1961. Huseyin K, Hagedorn P, Teschner W. On the stability of linear conservative gyroscopic systems. ZAMP 1983;34(6):807–15. Seyranian AP, Stoustrup J, Kliem W. On gyroscopic stabilization. ZAMP 1995;46:255–67. Koshlyakov VN. Structural transformations of dynamical systems with gyroscopic forces. J Appl Math Mech 1997;61(5):751–6. Merkin DR. Introduction to the Theory of Stability of Motion. Moscow: Nauka; 1976. Stoker JJ. Nonlinear Vibration in Mechanical and Electrical Systems. N.Y.: L. Interscience; 1950.
Translated by R.C.G.