Light damping

Solution If the eigenvalues are complex, the fixed point is either a center (Figure 5.2.4a) or a spiral (Figure 5.2.4b). We ve already seen an example of a center in the simple harmonic oscillator of Section 5.1 the origin is surrounded by a family of closed orbits. Note that centers are neutrally stable, since nearby trajectories are neither attracted to nor repelled from the fixed point. A spiral would occur if the harmonic oscillator were lightly damped. Then the trajectory would just fail to... 

Figure 8.5.10 explains why lightly damped Josephson junctions have hysteretic I — V curves. Suppose a is small and / is initially below the homoclinic bifurca-... 

Srinivasan, V., Schnitzlein, W. M., and Tripathy, D. K. (2001). Fowlpox encodes a novel repair enzyme, GPD-photolyase, that restores infectivity of UV light-damped vims. J. Virol. 75, 1681-1688. 

After the transformation of the response signal into the frequency domain, the damping ratio of a linear, viscously and lightly damped system can be determined using the bandwidth method given by... 

The plot of these functions, called the frequency response of the system, presents one of the most important features in forced vibration. In a lightly damped system, when the forcing frequency is close to the natural frequency (r 1), the amplitude of the vibration can get extremely large. This phenomenon is called resonance (consequently, the natural... 

Balas, G.J. Doyle, J.C. Control of lightly damped, flexible modes in the controller crossover region. J. of Guidance, Control Dynamics, 17, no. 2 (1994), pp. 370-377... 

When the vibrations are damped the amplitude decreases with time, but with light damping there is only a small effect on the period... 

The damping of excessive vibrations of as5un-metric plan multiple purpose buildings is a common task in civil engineering. Even discomfort of people living in lightly damped tall buildings is observed under wind excitation. Since the direct... 

Having evaluated the frequency response function F(v) of the deflection of the undamped plate, light damping is built in by an alternative quadrature type of the elastic-viscoelastic correspondence principle, /4/, /5/. 

For a lightly damped single-degree-of-freedom system, subjected to a narrow-band excitation at frequency (uf, the mean square system response is given by... 

The more traditional approach to estimate the modal parameters of a structure is often called peak picking method after its key step the identification of the resonant frequencies as the peaks of ASDs and CSDs. In fact, for a lightly damped structure subjected to a white-noise random excitation, both ASDs and CSDs reach a local maximum at the frequencies corresponding to the system normal modes (Bendat and Piersol 1993) furthermore, if the assumption is introduced of modes having well-separated frequencies, it can be shown (see, e.g., Peeters 2000) that... 

The method of stochastic averaging, firstly introduced by Landau and Stratonovich (1962), has proved to be a very useful tool for deriving approximate solutions to problems involving the dynamical response of lightly damped systems to broadband random excitation. It is based on the principle that the rate of change with respect to time of the oscillator s total energy is equal to the power input due to random excitation, minus the power dissipation due to the damping mechanism. 

In lightly damped structures the slowly varying energy can be treated as a constant over an appropriate period of oscillation, and oscillatory terms can be approximated by their time averages over one period of oscillation. Furthermore, under broadband random excitation, the relaxation time of the oscillator response is much greater than the correlation time of the excitation. Thus, it is possible to model the power input due to the excitation as a nonzero mean component plus an additional, fluctuating component with the character of white noise. 

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