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On the non-conservative nature of“gyroscopic conservative” systems
Journal of Sound and Vibration (1974) 36(3), 435--437
ON THE NON-CONSERVATIVE NATURE OF "GYROSCOPIC CONSERVATIVE" SYSTEMS The purpose of this note is to discuss the non-conservative nature of systems which have been designated "gyroscopic conservative" by Ziegler [I ]. He argued that the work done in a real motion by velocity dependent forces such as Coriolis and Lorentz forces and gyroscopic moments may be zero and hence that these "gyroscopic systems" are "obviously conservative 'a [1, p. 29]. As a result, the lowest critical load corresponds to a static (buckling) type of instability called divergence and may be determined by using a static equilibrium method. Nevertheless, Ziegler is careful to add that only a kinetic method of stability analysis will supply all the critical loads [1, p. 76]. In spite of this, the designation "conservative" to gyroscopic systems seems to have created considerable confusion and has led to incomplete analyses and erroneous conclusions as seen in references [2]-[6], to cite just a few. These papers all deal with the dynamics of tubes conveying fluids which, by virtue of the fluid Coriolis forces are So-called gyroscopic systems. The stability analyses have either been static or terminated when divergence was found, presumably because it was assumed that flutter could not occur. In each case, the authors have incorrectly concluded that tubes with both ends fixed (either clamped or pinned) are conservative and hence that flutter is impossible. Arguments have been based on examining the work done by the fluid during a cycle of tube oscillation: r
L
A W= - f MU[p z + Uyy']loL d t - 89 M [ p 2 - (Uy'Y]l'odx, 0
0
Where M is the mass of fluid per unit length of tube, U is the flow velocity~ and y is the transverse displacement. Clearly, for fixed ends, the first integral is always zero and, apparently, this is the only term considered in the above references. The second integral requires more careful examination. It appears that it will only be zero if the motion is periodic: i.e., if it is assumed apriori that the system is stable. The above expression for the work done is from the recent paper of Paidoussis and Issid [7] who re-examined the problem using a dynamic stability analysis. They established that coupled mode flutter in fact occurs for fixed-ended tubes at flow velocities above those required for divergence. Note that a typographical error appeared in this equation in reference [7] as was pointed out to the author in a private communication with Professor Paidoussis. The sign of the second term in the second integral must be negative in order that, for sufficiently high flow velocities, the work done is positive and oscillations of increasing amplitude will occur. It is now known that all finite, fixed-ended plates, shells and pipes in a subsonic flow behave in a similar manner with static divergence preceding flutter. Kornecki [8] has shown this through an interesting qualitative examination of the fluid forces. In a quantitative analysis, Paidoussis [9] has demonstrated the equivalence of long shells and tubes with an internal flow. Huseyin and Plaut [I0] have recently developed a general theory for a class of linear "gyroscopic conservative" systems with special emphasis on flutter instability. They show that these systems may be expressed in the form (specialized here for fluid flow problems) {-o92[m] + io9[g] + [k] - U2[be]}y
=
0,
where o9is the complex frequency, [m] is a positive definite mass matrix including virtual mass, i = (-1) 112, U is the flow velocity, [g] is a skew symmetric matrix due to Coriolis forces, [k] is 435
436
LETTERSTO THE EDITOR
a positive definite matrix due to elastic restoring forces and [bc] is a positive definite matrix which depends on the boundary conditions. One of the examples used by Huseyin and Plaut to illustrate their theory and demonstrate the existence of flutter is a Galerkin solution of a fixed-ended tube conveying fluid. It is easily shown that the plate and shell analyses of Weaver and Unny [11, 12] can be expressed in the same form and hence that they belong to the same class. It was found in those papers that flutter followed divergence. In spite of the fact that it has been proven that coupled mode flutter exists in these systems, the designation "gyroscopic conservative" seems to have persisted. This behaviour can only be the result of non-conservative forces transferring energy to the elastic structure. For.fluid flow problems, these forces are the aero- or hydrodynamic forces which clearly depend on the motion of the structure. Mathematically, this is reflected in the non-selfadjoint nature of the differential equations of motion. Such equations admit complex eigenvalues or, more relevant to the present discussion, the complex frequencies characteristic of flutter. It is important to add that the eigenvalues need not always be complex, as is the case for flow velocities below the critical flutter velocity. However, this special case does not change the basic nature of the system and justify calling it conservative. It simply implies that a non-conservative system may sometimes behave like a conservative one. Von Karman and Biot [13] recognized that gyroscopic forces could not be derived from the potential energy of a system and called them non-conservative. Leipholz [14] has called such systems slightly asymmetric. Whatever they are called, in the light of our present knowledge it does not seem consistent physically or mathematically to continue to refer to them as conservative.
ACKNOWLEDGMENT The author would like to express his gratitude to K. Huseyin and M. P. Paidoussis for making their recent papers available to him prior to their publication. Department of 2~lecha/zical Engineering, McMaster University, Hamilton, Ontario, Canada L8S 4L7 (Receh'ed 7 May 1974)
D.S. WEAVER
REFERENCES 1. H. ZIE~LER1968 Principles of Structural Stability, Waltham, Massachusetts: Blaisdell Publishing Company. 2. M. P. PAIDOUSSIS1970 Journal of Mechanical Engineerhlg Science 12, 85-103. Dynamics of tubular cantilevers conveying fluid. 3. S. S. CHEN 1971 American Society of ~[echanieal Engineers Vibrations Conference, Toronto, Canada, paper 71-FIBR-39. Flow-induced instability of an elastic tube. 4. S. S. CtlEN 1971 American Society of Civil Enghwers Journal of the Engbteering Mechanics Dit'ision 97, 1469-1485. Dynamic stability of tube conveying fluid. 5. M. P. PAiooossls 1972 Proceedings of lUTAM/1AtlR Symposhtm on Flow-hi&wed Structural Vibrations, Karlsrtthe paper G-3. Stability of tubular cylinders conveying fluid. 6. S. S. CHEN 1973 American Society of AIechanical Engineers Journal of Applied Mechanics 40, 362-368. Out-of-plane vibration and stability of curved tubes conveying fluid. 7 M.P. PAioousszs and N. T. lsslo 1974 Journal of Sound and Vibration 33, 267-294. Dynamic stability of pipes conveying fluid. 8. A. KORNEr 1974 Journal of Sound and Vibration 32, 251-263. Static and dynamic instability of panels and cylindrical shells in subsonic potential flow. 9. M. P. PAlooffssls 1973 McGill Uni~'ersity, Department of Mechanical Engineering, MERL TN 73-2. Flutter of conservative systems of pipes containing incompressible flow.
LETTERS TO THE EDITOR
437
10. K. HUSEYIN and R. H. PLAUT 1974 (to appear) JOltrnal of Structural Mechanics. Transverse vibrations and stability of gyroscopic systems. 11. D. S. WEAVERand T. E. Ur~JY 1970 Journal of,4pplied ~1echanics 37, 823-826. The hydroelastie stability of a flat plate. 12. D. S. WEAVER and T. E. UNNY 1973 Journal of,4pplied Mechanics 40, 48-53. On the dynamic stability of fluid conveying pipes. 13. T. YON KARMAN and M. A. BIOT 1940 Mathematical Methods hz Engineering. New York: McGraw-Hill Book Company, Inc. 14. H. H. E. LEIPHOLZ 1970 Stability Theory. New York: Academic Press.
ON THE NON-CONSERVATIVE NATURE OF "GYROSCOPIC CONSERVATIVE" SYSTEMS The purpose of this note is to discuss the non-conservative nature of systems which have been designated "gyroscopic conservative" by Ziegler [I ]. He argued that the work done in a real motion by velocity dependent forces such as Coriolis and Lorentz forces and gyroscopic moments may be zero and hence that these "gyroscopic systems" are "obviously conservative 'a [1, p. 29]. As a result, the lowest critical load corresponds to a static (buckling) type of instability called divergence and may be determined by using a static equilibrium method. Nevertheless, Ziegler is careful to add that only a kinetic method of stability analysis will supply all the critical loads [1, p. 76]. In spite of this, the designation "conservative" to gyroscopic systems seems to have created considerable confusion and has led to incomplete analyses and erroneous conclusions as seen in references [2]-[6], to cite just a few. These papers all deal with the dynamics of tubes conveying fluids which, by virtue of the fluid Coriolis forces are So-called gyroscopic systems. The stability analyses have either been static or terminated when divergence was found, presumably because it was assumed that flutter could not occur. In each case, the authors have incorrectly concluded that tubes with both ends fixed (either clamped or pinned) are conservative and hence that flutter is impossible. Arguments have been based on examining the work done by the fluid during a cycle of tube oscillation: r
L
A W= - f MU[p z + Uyy']loL d t - 89 M [ p 2 - (Uy'Y]l'odx, 0
0
Where M is the mass of fluid per unit length of tube, U is the flow velocity~ and y is the transverse displacement. Clearly, for fixed ends, the first integral is always zero and, apparently, this is the only term considered in the above references. The second integral requires more careful examination. It appears that it will only be zero if the motion is periodic: i.e., if it is assumed apriori that the system is stable. The above expression for the work done is from the recent paper of Paidoussis and Issid [7] who re-examined the problem using a dynamic stability analysis. They established that coupled mode flutter in fact occurs for fixed-ended tubes at flow velocities above those required for divergence. Note that a typographical error appeared in this equation in reference [7] as was pointed out to the author in a private communication with Professor Paidoussis. The sign of the second term in the second integral must be negative in order that, for sufficiently high flow velocities, the work done is positive and oscillations of increasing amplitude will occur. It is now known that all finite, fixed-ended plates, shells and pipes in a subsonic flow behave in a similar manner with static divergence preceding flutter. Kornecki [8] has shown this through an interesting qualitative examination of the fluid forces. In a quantitative analysis, Paidoussis [9] has demonstrated the equivalence of long shells and tubes with an internal flow. Huseyin and Plaut [I0] have recently developed a general theory for a class of linear "gyroscopic conservative" systems with special emphasis on flutter instability. They show that these systems may be expressed in the form (specialized here for fluid flow problems) {-o92[m] + io9[g] + [k] - U2[be]}y
=
0,
where o9is the complex frequency, [m] is a positive definite mass matrix including virtual mass, i = (-1) 112, U is the flow velocity, [g] is a skew symmetric matrix due to Coriolis forces, [k] is 435
436
LETTERSTO THE EDITOR
a positive definite matrix due to elastic restoring forces and [bc] is a positive definite matrix which depends on the boundary conditions. One of the examples used by Huseyin and Plaut to illustrate their theory and demonstrate the existence of flutter is a Galerkin solution of a fixed-ended tube conveying fluid. It is easily shown that the plate and shell analyses of Weaver and Unny [11, 12] can be expressed in the same form and hence that they belong to the same class. It was found in those papers that flutter followed divergence. In spite of the fact that it has been proven that coupled mode flutter exists in these systems, the designation "gyroscopic conservative" seems to have persisted. This behaviour can only be the result of non-conservative forces transferring energy to the elastic structure. For.fluid flow problems, these forces are the aero- or hydrodynamic forces which clearly depend on the motion of the structure. Mathematically, this is reflected in the non-selfadjoint nature of the differential equations of motion. Such equations admit complex eigenvalues or, more relevant to the present discussion, the complex frequencies characteristic of flutter. It is important to add that the eigenvalues need not always be complex, as is the case for flow velocities below the critical flutter velocity. However, this special case does not change the basic nature of the system and justify calling it conservative. It simply implies that a non-conservative system may sometimes behave like a conservative one. Von Karman and Biot [13] recognized that gyroscopic forces could not be derived from the potential energy of a system and called them non-conservative. Leipholz [14] has called such systems slightly asymmetric. Whatever they are called, in the light of our present knowledge it does not seem consistent physically or mathematically to continue to refer to them as conservative.
ACKNOWLEDGMENT The author would like to express his gratitude to K. Huseyin and M. P. Paidoussis for making their recent papers available to him prior to their publication. Department of 2~lecha/zical Engineering, McMaster University, Hamilton, Ontario, Canada L8S 4L7 (Receh'ed 7 May 1974)
D.S. WEAVER
REFERENCES 1. H. ZIE~LER1968 Principles of Structural Stability, Waltham, Massachusetts: Blaisdell Publishing Company. 2. M. P. PAIDOUSSIS1970 Journal of Mechanical Engineerhlg Science 12, 85-103. Dynamics of tubular cantilevers conveying fluid. 3. S. S. CHEN 1971 American Society of ~[echanieal Engineers Vibrations Conference, Toronto, Canada, paper 71-FIBR-39. Flow-induced instability of an elastic tube. 4. S. S. CtlEN 1971 American Society of Civil Enghwers Journal of the Engbteering Mechanics Dit'ision 97, 1469-1485. Dynamic stability of tube conveying fluid. 5. M. P. PAiooossls 1972 Proceedings of lUTAM/1AtlR Symposhtm on Flow-hi&wed Structural Vibrations, Karlsrtthe paper G-3. Stability of tubular cylinders conveying fluid. 6. S. S. CHEN 1973 American Society of AIechanical Engineers Journal of Applied Mechanics 40, 362-368. Out-of-plane vibration and stability of curved tubes conveying fluid. 7 M.P. PAioousszs and N. T. lsslo 1974 Journal of Sound and Vibration 33, 267-294. Dynamic stability of pipes conveying fluid. 8. A. KORNEr 1974 Journal of Sound and Vibration 32, 251-263. Static and dynamic instability of panels and cylindrical shells in subsonic potential flow. 9. M. P. PAlooffssls 1973 McGill Uni~'ersity, Department of Mechanical Engineering, MERL TN 73-2. Flutter of conservative systems of pipes containing incompressible flow.
LETTERS TO THE EDITOR
437
10. K. HUSEYIN and R. H. PLAUT 1974 (to appear) JOltrnal of Structural Mechanics. Transverse vibrations and stability of gyroscopic systems. 11. D. S. WEAVERand T. E. Ur~JY 1970 Journal of,4pplied ~1echanics 37, 823-826. The hydroelastie stability of a flat plate. 12. D. S. WEAVER and T. E. UNNY 1973 Journal of,4pplied Mechanics 40, 48-53. On the dynamic stability of fluid conveying pipes. 13. T. YON KARMAN and M. A. BIOT 1940 Mathematical Methods hz Engineering. New York: McGraw-Hill Book Company, Inc. 14. H. H. E. LEIPHOLZ 1970 Stability Theory. New York: Academic Press.