Two degree of freedom gyroscopic systems with linear integrals

In this paper we consider the first integrals, linear in velocities, of conservative gyroscopic systems with two degrees of freedom. A precise criterion which specifies whether a given gyroscopic system possesses a linear integral is derived. When the kinetic energy has the structure of a standard form of the metric of revolution, all the possible systems which admit a linear integral and corresponding integrals are determined explicitly. Two examples are considered to illustrate the usefulness of the derived results.     

Coexistence phenomenon in autoparametric excitation of two degree of freedom systems

Coexistence phenomenon refers to the absence of expected tongues of instability in parametrically excited systems. In this paper we obtain sufficient conditions for coexistence to occur in the generalized Ince equation

     

On the instability of parametrically excited two degrees of freedom vibrating systems with viscous damping

Summary The stability of vibrating systems having two degrees of freedom subjected to parametric excitation and with viscous damping are studied with a special perturbation method.Analytical expressions in explicit form in second approximation are obtained for the transition from stability to instability. A general criterion for judging the limits within which the approximation is acceptable is indicated.It is shown that even in parametrically excited vibrating systems damping can have a destabilising effect similar to the destabilising effect known to exist in nonconservative elastic systems.

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Sommario Nel presente lavoro si esamina con un particolare metodo di perturbazione la stabilità dei sistemi vibranti a due gradi di libertà eccitati parametricamente e con resistenza viscosa.Si ottengono espressioni analitiche in forma esplicita in seconda approssimazione dei confini fra stabilità ed instabilità. Si dà anche un criterio di massima per giudicare entro quali limiti l'approssimazione risulta accettabile.Si pone in evidenza come anche nelle vibrazioni eccitate parametricamente lo smorzamento possa avere un effetto instabilizzante analogo a quello già noto nei sistemi elastici non conservativi.

     

The geometrical stability of non-linear normal modes in two degree of freedom systems

The geometrical stability of the non-linear normal mode vibrations of a class of two degree of freedom dynamical systems is studied by utilizing the definitions and analysis of Synge's “Geometry of Dynamics.”It is shown that instabilities can occur for small amplitudes of vibration only if (a) one of the associated linear normal modes possesses a frequency which is nearly a multiple of the frequency of the other linear normal mode, or (b) the frequency of one linear normal mode is nearly zero.     

Adiabatic invariants for dynamical systems with one degree of freedom

Adiabatic invariants for dynamical systems with one degree of freedom, whose equation of motion is (1), and where the existence of the corresponding Hamilton action integral is not imposed, are established. The adiabatic invariants may vary according to their structure. Using the theory a few particular problems, including non-autonomous Duffing and Van der Pol oscillators, are analysed. Finally, it is indicated how the adiabatic invariants can be used for finding approximate solutions, stability analyses and for plotting phase curves.     

Subharmonic steady vibrations of a non-linear damper with two degrees of freedom and viscous damping

The steady-state. $\text{1}\text{3}$ subharmonic vibrations of a dynamic damper (or vibration absorber) with two degrees of freedom, sinusoidal forcing function and internal viscous damping. are presented. The study of these oscillations leads to the determination of suitable “form functions” of the solutions, by following a methodology recently introduced by Nocilla for studying the harmonic vibrations of non-linear systems with one and two degrees of freedom. The proposed theory. which is valid even if the non-linearity is large, gives satisfactory results in all the cases in which the subharmonic component is predominant in the steady-state oscillation of the system.     

Vibrational control of mechanical systems with piecewise linear damping and high-frequency inputs

Nonlinear Dynamics - This paper discusses averaging and vibrational control of mechanical control-affine systems with piecewise linear damping and high-frequency inputs. The results are used for...     

Modal interaction and bifurcations in two degree of freedom duffing oscillators

A methodology is first presented for analyzing long time response of periodically exited nonlinear oscillators. Namely, a systematic procedure is employed for determining periodic steady state response, including harmonic and superharmonic components. The stability analysis of the located periodic motions is also performed, utilizing results of Froquet theory. This methodology is then applied to a special class of two degree of freedom nonlinear oscillators, subjected to harmonic excitation. The numberical results presented in the second part of this study illustrate effects caused by the interaction of the modes as well as effects of the nonlinearities on the steady state response of these oscillators. In addition, sequences of bifurcations are analyzed for softening systems, leading to unbounded response of the model examined. Finally, the importance of higher harmonics on the response of systems with strongly nonlinear characteristics is investigated.     

Localization of energy in nonlinear systems with two degrees of freedom

Energy pumping in a two-degrees-of-freedom system with linear and essentially nonlinear oscillators is studied. The kinetic energy envelopes of the linear and nonlinear subsystems are chosen to be the main characteristics of the process under consideration. A criterion that the energy of the linear oscillator excited at time zero is completely pumped over into the nonlinear oscillator is established together with an additional condition whereby the energy does not return to the linear subsystem. Optimal energy pumping mode is established using global optimization. The effect of the parameters of the system on the main characteristics is assessed __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 5, pp. 115–125, May 2007.     

The stability boundary of systems with one degree of freedom (Part II)

Summary This paper is designed as a supplement to Part I with a view to adding to the practical value of the theory presented there. Singular critical points which may arise under certain circumstances, and the effect of their presence on the shape of the stability boundary are investigated. It is shown that such points, if they exist, will generally lie on planes in the loading space, and that a simple linear eigenvalue approach will readily yield their location. An example is solved for illustration, and the results of an experiment are reported.

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Sommario Questa nota rappresenta un'integrazione ad una precedente (Parte I) e si propone di accrescere il valore applicativo della teoria ivi sviluppata. Si studiano punti critici singolari che possono presentarsi in certe circostanze, e le conseguenze della loro presenza sulla forma del dominio di stabilità. Si dimostra che se tali punti esistono, appartengono in generale a piani nello spazio dei carichi e che la loro ubicazione può essere facilmente determinata con un semplice procedimento fondato sulla considerazione degli autovalori. La risoluzione di un esempio e risultati sperimentali illustrano le conclusioni raggiunte in sede teorica.

     

Nearly hamiltonian systems in two degrees of freedom

Perturbed systems of differential equations are studied which are close to Hamiltonian systems of the type for which small divisor problems arise. If the non-Hamiltonian terms involve both damping and self-excitation, behavior characteristic of Hamiltonian systems can occur but is confined to small regions of space in which these opposite effects are in balance. The results provide a partial justification for truncating the Fourier series occurring in these problems. The principle method is local averaging, which is used to obtain global results by compactness arguments.     

Multiple parametric resonance in a non-linear two degree of freedom system

A non-linear two degrees of freedom dynamical system subjected to parametric excitation is studied. The relationship between the parametric frequency and the natural frequencies of the system are such that the first generalized coordinate is excited into parametric resonance while both generalized coordinates are excited into combination resonance at the same time. The unstable zone and response of the system under multi-resonant condition are then compared with the unstable zones and responses of the system where each type of resonance is treated on an individual basis, assuming the other type of resonance does not occur.     

Some observations on two piecewise-linear dynamic systems with induced hysteretic damping

Significant hysteretic damping can be introduced into dynamical systems for vibration control purpose by means of semi-active control implementation. The dynamic behavior of two such models with a single-degree-of-freedom are examined in this paper. Control performance is found to be not only dependent on the stiffness ratio between the primary spring and auxiliary spring, but also on the forcing frequency of the harmonic excitation. Some problems associated with the equivalent linearization of these types of dynamical systems are also discussed.     

A note on stabilizing damping configurations for linear nonconservative systems

It is well known that a linear elastic system having a stable equilibrium for zero damping may be destabilized by the introduction of nonzero damping. In this note a class of damping configurations not having this destabilizing effect is determined.