Nonconservative Problems Research Articles

On variational methods for nonconservative problems

On variational methods for nonconservative problems

On non-conservative stability problems of rotating elastic systems with internal damping

The state of stability of a uniformly rotating double pendulum carrying a tip mass,having viscoelastic joints, and subjected to a tangential force applied at its free end is investigated. The system is assumed to rotate at constant angular velocity about either its own axis or an axis perpendicular to the plane of oscillations. Generic relationships between critical loads for no viscous damping in the hinges and for slight damping as well as vanishing damping are derived. Loss of stability is observed to occur through divergence of flutter for compressive and, under certain circumstances, tensile tangential loads depending upon the values of the mass ratio, damping coefficient ratio, and angular velocity parameters.

Ritz method in non-conservative instability problems: A simple example

The immediate post-critical behaviour of a cantilever column with subtangential follower force loading is studied using a Ritz method to discretize the problem. The example is intended to show that the Ritz method can be a useful technique in the analysis of practical problems of the dynamic stability of elastic systems with non-conservative loading. The type of instability exhibited by the cantilever is shown to depend on the slope ratio of the subtangential follower force, with the possibility of either hard or soft flutter instability occurring in different ranges, as well as static buckling.

Conservation laws in classical mechanics for quasi-coordinates

Noether's theorem and Noether's inverse theorem for mechanical systems with gauge-variant Lagrangians under symmetric infinitesimal transformations and whose motion is described by quasi-coordinates are established. The existence of first integrals depends on the existence of solutions of the system of partial differential equations — the so-called Killing equations. Non-holonomic mechanical systems are analysed separately and their special properties are pointed out. By use of this theory, the transformation which corresponds to Ko Valevskaya first integral in rigid-body dynamics is found. Also, the nature of the energy integral in non-holonomic mechanics is shown and a few new first integrals for non-conservative problems are obtained. Finally, these integrals are used in constructing Lyapunov's function and in the stability analyses of nonautonomous systems. The theory is based on the concept of a mechanical system, but the results obtained can be applied to all problems in mathematical physics admitting a Lagrangian function.

On the method of Trefftz in the light of a generalized variational principle

An extended version of the Principle of Virtual Work is being used to derive Trefftz' equations. This new approach is superior to the classical one, since it allows to include nonconservative problems. In the case of such problems involving nonconservative forces, one may have to use so called Quasi-Trefftz' equations instead of the simpler Trefftz' equations. In order to show the applicability of the Quasi-Trefftz equations, a proof of convergence is given for a rather general example.

On the derivation of Leipholz's variational principle for non-conservative stability problems

On the derivation of Leipholz's variational principle for non-conservative stability problems

A nonconservative stability problem with unstable positions of equilibrium for arbitrarily small loads

A nonconservative stability problem with unstable positions of equilibrium for arbitrarily small loads

Comments on the paper: Adjoint variational methods in nonconservative stability problems [1

Comments on the paper: Adjoint variational methods in nonconservative stability problems [1

Application of a variational method to dissipative, non-conservative problems of elastic stability

This study deals with the development of an adjoint variational principle that forms the basis of a method that is used to obtain approximate solutions for non-conservative problems of elastic stability in which dissipative forces are present. For a general fifth-order partial differential equation in time and one space variable, the associated adjoint boundary value problem is derived. A variational principle embodying both the original and the adjoint problems is developed. Three specific non-conservative stability problems are studied by this method and the numerical convergence of the approximate solutions is studied by enlarging suitably the number of modes retained in the assumed expansions of the deflection functions. It is found that internal damping may be either of a stabilizing or destabilizing nature, depending upon its magnitude as well as the magnitude of the external damping parameter.

Application of the complementary energy method to two non-conservative problems of elastic stability

The methods of Galerkin and complementary energy in conjunction with an adjoint variational principle are applied to two non-conservative problems of elastic stability. These problems consist of (i) a cantilever beam and (ii) a clamped-hinged beam subjected to a linearly distributed tangential compressive load. The numerical results indicate that, in these examples, the method of complementary energy converges somewhat faster than the Galerkin method; however the former requires a little more effort to apply. In addition, the numerical computations lead to values of the critical load intensity parameter that are 5% and 28% lower, respectively, than previously published values.

On the role of the adjoint problem in dissipative, nonconservative problems of elastic stability

The asymptotic stability of linear isotropic Kelvin-Voigt solids subjected to external damping and noncon-servative surface tractions which are linear functions of displacements and displacement gradients is considered. The boundary value problem that is adjoint to the original system is derived, and a variational principle from which these two boundary value problems may be generated is stated. The variational principle serves as a basis for an approximate method (similar to the Ritz method) for solving nonconservative stability problems in which the effects of internal and external damping are present. The method is applied to determine the value of the critical load intensity Qcr in a cantilever and a clamped-simply supported beam subjected to a linearly distributed tangential load as well as to internal and external damping forces. Plots of the variation of Qcr with the two damping parameters are given, and it is shown the nature of the boundary conditions can have a significant effect on the manner in which Qcr varies as a function of the damping parameters.

Determining the Nature of Instability in Nonconservative Problems

Conclusions Wind-tunnel Magnus data obtained to date on canted fin configurations may be invalid in that a severe normal force interaction may be present. Although it may be possible in some cases to correct for this interaction, it is in general best to remove the interaction by subtracting the zero spin data at each angle of attack. This requires a system in which the model can be kept at zero spin during an angle of attack sweep of the configuration.

A comparison of approximate methods for solving non-conservative problems of elastic stability

The methods of Ritz, Galerkin, and complementary energy are applied to a nonconservative problem in the theory of elastic stability. The numerical calculations are based upon (i) a variational expression, for which no functional can be determined, and (ii) an adjoint variational principle, for which a functional is established in terms of the variables of the original non-self-adjoint eigenvalue problem and the adjoint problem. The adjoint variational principle yields somewhat more accurate values for the critical load parameter than does the variational expression. In addition, the results obtained by means of the complementary energy method are more precise than the corresponding results obtained from the Ritz and Galerkin methods.

Adjoint variational methods in nonconservative stability problems

A general nonself-adjoint eigenvalue problem is examined and it is shown that the commonly employed approximate methods, such as the Galerkin procedure, the method of weighted residuals and the least square technique lack variational descriptions. When used in their previously known forms they do not yield stationary eigenvalues and eigenfunctions. With the help of an adjoint system, however, several analogous variational descriptions may be developed and it is shown in the present study that by properly restating the method of least squares, stationary eigenvalues may be obtained. Several properties of the adjoint eigenvalue problem, known only for a restricted group, are shown to exist for the more general class selected for study.

Hypersonic Turbulent Skin-Friction and Boundary-Layer Profiles on Nonadiabatic Flat Plates

Random Processes, Journal of the Acoustical Society of America, Vol. 49, No. 1, Jan. 1971, pp. 556-583. 3 Bolotin, V. V., Nonconservative Problem of the Theory of Elastic Stability, Pergamon Press, New York, 1963, pp. 280-285. 4 Chu, H. and Herrman, G., Influence of Large Amplitude on Free Flexural Vibration of Rectangular Elastic Plates, Journal of Applied Mechanics, Vol. 23, No. 4, Dec. 1956, pp. 532-540. 5 Bull, M. K., Wall-Pressure Fluctuation Associated with Subsonic Turbulent Boundary Layer Flow, Journal of Fluid Mechanics, Vol. 28, Pt. 4,1967, pp. 719-754. 6 Bull, M. K., Wilby, J. F., and Blackman, D. R., Wall Pressure Fluctuation in Boundary Layer Flow and Response of Simple Structure to Random Pressure Fields, A.A.S.U. Rept. 243, Univ. of Southampton, 1963, pp. 1-28. 7 Strawderman, W. A., Turbulent-Induced Plate Vibrations, An Evaluation of Finite and Infinite Plate Models, Journal of the Acoustical Society of America, 1969, pp. 1294-1307. 8 Maestrello, L., Measurement and Analysis of the Response Field of Turbulent Boundary Layer Excited Panels, Journal of Sound and Vibration, Vol. 2, 1965, pp. 270-292.

Nonconservative problem concerning the oscillations of a solid body containing a cavity partially filled with an ideal liquid

Nonconservative problem concerning the oscillations of a solid body containing a cavity partially filled with an ideal liquid

Approaches to the Nonconservative Problems of Instability of Columns by means of Difference Method

Approaches to the Nonconservative Problems of Instability of Columns by means of Difference Method

Studies on Nonconservative Problems of Instabilities of Columns by means of Analog Computer

Studies on Nonconservative Problems of Instabilities of Columns by means of Analog Computer

Some Observations on Nonconservative Problems of Elastic Stability

With reference to undamped elastic systems subjected to nonconservative forces, the possibility of a transition from stability to divergence through infinitely large values of the frequency is considered. Examined in detail is the special case of a cantilever column with two concentrated masses and subjected to a follower end force. It is shown that either flutter or divergence may first set in, depending on the relative magnitudes and locations of the masses. The results obtained by letting one of the masses tend to zero are not necessarily consistent with those obtained directly by setting that mass equal to zero; alternately, it may be stated that an additional infinitesimal mass may lead to drastic changes in the stability behavior of a given elastic system subjected to nonconservative forces.

Variational Method for Nonconservative Problems

It is shown that the Ritz variational method can be applied to nonconservative problems provided the admissible velocity satisfies not only the geometric boundary condition but also a work condition on the part of the boundary surface where traction and traction rate are prescribed. It is further found that the modified variational principle can also be applied to nonconservative systems for which it is possible to construct an adjoint system. The principle is expected to be useful for application to a wide class of problems, including those encountered in connection with hydrodynamic and hydromagnetic stability investigation.