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.

     

Random vibrations of a rotating shaft with non-linear damping

Bending vibrations of a rotating shaft due to external random excitation are considered for the case of potential instability of the shaft's linear model due to the presence of internal or “rotating” damping. A two-degree-of-freedom model is studied which accounts for non-linearity in external or “non-rotating” damping. An explicit expression is obtained for a stationary joint probability density of displacements and velocities as an exact analytical solution to the corresponding Fokker-Planck-Kolmogorov equation. The results are used to develop criterion for on-line detection of instability for the operating shaft from its measured response.     

Subharmonic solution of a piecewise linear oscillator with tow degrees of freedom

In the present paper subharmonic resonance solution ofapiecewise linear oscillator with two degrees of freedom is studied. It is shown that in this system there exist a series of subharmonic resonance solutions, among them there are 1/2, 1/3, 1/4, 1/5, 1J6,……subharmonic resonance solutions. The calculated results by the analogy computer and the field experiments in the factory partly verify this theory. Under certain circumstances, the generation of chaotic states of the oscillation is observed in analogy computer solutions.     

Reducing force transmissibility in multiple degrees of freedom structures through anti-symmetric nonlinear viscous damping

In the present study, the Volterra series theory is adopted to theoretically investigate the force transmissibility of multiple degrees of freedom (MDOF) structures, in which an isolator with nonlinear anti-symmetric viscous damping is assembled. The results reveal that the anti-symmetric nonlinear viscous damping can significantly reduce the force transmissibility over all resonance regions for MDOF structures with little effect on the transmissibility over non-resonant and isolation regions. The results indicate that the vibration isolators with an anti-symmetric damping characteristic have great potential to solve the dilemma occurring in the design of linear viscously damped vibration isolators where an increase of the damping level reduces the force transmissibility over resonant frequencies but increases the transmissibility over non-resonant frequency regions. This work is an extension of a previous study in which MDOF structures installed on the mount through an isolator with cubic nonlinear damping are considered. The theoretical analysis results are also verified by simulation studies.     

The effects of internal resonance on impulsively forced non-linear systems with two degrees of freedom

The method of multiple time scales is used to study the non-linear oscillations of impulsively forced systems under conditions of internal resonance. A partial analytical solution is obtained.The method is illustrated by an example in which the internal resonance effects are shown to be significant.     

The force transmissibility of MDOF structures with a non-linear viscous damping device

In the present study, the concept of the Output Frequency Response Function (OFRF), recently proposed by the authors, is applied to theoretically investigate the force transmissibility of MDOF structures with a cubic non-linear viscous damping device. The results analytically show that the introduction of cubic non-linear damping can significantly reduce the transmissibility over all resonance regions for a Multiple Degree of Freedom (MDOF) structure and at the same time leave the transmissibility over the isolation region virtually unaffected. The analysis also indicates that a strong linear damping may shift the system resonances and compromise the beneficial effects of cubic non-linear viscous damping on the force transmissibility of MDOF structures. This suggests that a less significant linear damping together with a strong cubic non-linear damping can be used in MDOF structures to achieve a desired vibration isolation performance. This research work has a significant implication for the design of viscously damped MDOF structures for a wide range of practical applications.     

On damped non-linear dynamic systems with many degrees of freedom

Non-linear mass-spring-damper systems with many degrees of freedom are studied; all springs and/or all dampers may be strongly non-linear. It is shown that the ultimate state of completely damped systems is always rest, that of incompletely damped systems may be either rest or a periodic normal mode motion. Necessary and sufficient conditions are given for the existence of classical normal mode motion in completely or incompletely damped systems. When the system is linear, these reduce to known results found by Rayleigh and generalized by Caughey and O'Kelly.     

Forced oscillations of a piecewise-linear non-linear dynamic system with several degrees of freedom

Exact periodic solutions are derived for a dynamic system with several degrees of freedom consisting of a series of ‘Reid springs’ with piecewise-linear, non-linear characteristis; however, the solutions are restricted to a class of harmonic excitation in the ‘modal form’ described subsequently in the paper. Conditions are derived for the asymptotic stability of the periodic solution and an example has been worked out in detail on the response of a dynamic system with two degrees of freedom.     

Approximate backbone curves for non-linear systems with several degrees of freedom

The concept or forced normal modes for linear damped systems is extended to non-linear systems. It is used together with the first harmonic method in order to establish an approximate but rapid method for the determination of backbone curves for non-linear systems having several degrees of freedom.     

Transient process of the vibrations of a cylindrical shell as a system with many degrees of freedom

A procedure is developed for investigating the transient process associated with the forced vibrations of a cylindrical shell as a system with many degrees of freedom. The amplitudes of the bending deflections of circular modes induced by a driving load of variable frequency and amplitude are compared with the same amplitudes for a uniform driving load and steady-state vibrations. The total maximum deflections of all the investigated cases are found to differ substantially. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 3, pp. 57–63, March, 1999.     

Detailed asymptotics for a convex Hamiltonian system with two degrees of freedom

The system of ordinary differential equations

A quasi-zero-stiffness isolator with a shear-thinning viscous damper

Quasi-zero-stiffness(QZS) vibration isolators have been widely studied,because they show excellent high static and low dynamic stiffnesses and can effectively solve low-frequency and ultralow-frequency vibration. However, traditional QZS(T-QZS)vibration isolators usually adopt linear damping, owing to which achieving good isolation performance at both low and high frequencies is difficult. T-QZS isolators exhibit hardening stiffness characteristics, and their vibration isolation performance is e...     

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.     

Wake-induced vibration of tandem and staggered cylinders with two degrees of freedom

The wake-induced vibration (WIV) of two staggered cylinder with two degrees of freedom (2-dof) has been investigated by experiments in a water channel for Reynolds number between 2000 and 25 000. The streamwise separation was fixed to 4 diameters and the lateral separation varied between 0 and 3 diameters for tandem and staggered configurations. Results are presented in the form of trajectories of motion and dynamic response curves of displacements, frequencies and force coefficients. Excitation caused by the WIV mechanism is found to get weaker as the initial position of the downstream cylinder is increased from the centreline of the wake (tandem arrangement) towards the sides. For a lateral separation of 3 diameters wake interference was already found to be negligible. Evidence of a type of wake-stiffness concept is also observed to occur for 2-dof WIV in tandem arrangement, especially for higher reduced velocities. A similar mechanism may also be occurring for staggered arrangements around the centreline.