Transient and chaotic low-energy transfers in a system with bistable nonlinearity

The low-energy dynamics of a two-dof system composed of a grounded linear oscillator coupled to a lightweight mass by means of a spring with both cubic nonlinear and negative linear components is investigated. The mechanisms leading to intense energy exchanges between the linear oscillator, excited by a low-energy impulse, and the nonlinear attachment are addressed. For lightly damped systems, it is shown that two main mechanisms arise: Aperiodic alternating in-well and cross-well oscillations of the nonlinear attachment, and secondary nonlinear beats occurring once the dynamics evolves solely in-well. The description of the former dissipative phenomenon is provided in a two-dimensional projection of the phase space, where transitions between in-well and cross-well oscillations are associated with sequences of crossings across a pseudo-separatrix. Whereas the second mechanism is described in terms of secondary limiting phase trajectories of the nonlinear attachment under certain resonance conditions. The analytical treatment of the two aformentioned low-energy transfer mechanisms relies on the reduction of the nonlinear dynamics and consequent analysis of the reduced dynamics by asymptotic techniques. Direct numerical simulations fully validate our analytical predictions

On the periodic motions of simple hopping robots

Discrete dynamical systems theory is applied to the analysis of simplified hopping robot models. A one-dimensional vertical hopping model that captures both the vertical hopping dynamics and nonlinear control algorithm is reviewed. A more complicated two-dimensional model that includes both forward and vertical hopping dynamics and a foot placement algorithm is presented. These systems are analyzed using a Poincare return map and hopping behavior is investigated by constructing the return map bifurcation diagrams with respect to system parameters. The diagrams show period doubling leading to chaotic behavior. Using the vertical model results as a guide, dynamic behaviour of the planar hopping system is interpreted

Energy equipartition in two-dimensional granular systems with spherical intruders

We study the effects of a line of spherical interstitial particles (or intruders) placed between two adjacent uncompressed chains of larger particles in a square packing of spheres, using experiments and numerical simulations. We excite one of the chains of particles adjacent to the intruders with an impact and show how energy is transmitted across the system until equipartition is reached from the excited (or impacted) chain to the absorbing (or adjacent) chain. The coupling of the two chains, although a purely two-dimensional effect, is modeled by a simplified one-and-a-half-dimensional system in which transverse motions of the particles are neglected

Discrete breathers at the interface between a diatomic and monoatomic granular chain

In the present work, we develop a systematic examination of the existence, stability and dynamical properties of a discrete breather at the interface between a diatomic and a monoatomic granular chain. We remarkably find that such an "interface breather" is more robust than its bulk diatomic counterpart throughout the gap of the linear spectrum. The latter linear spectral gap needs to exist for the breather state to arise and the relevant spectral conditions are discussed. We illustrate the minimal excitation conditions under which such an interface breather can be "nucleated" and analyze its apparently weak interaction with regular highly nonlinear solitary waveforms.Comment: 11 pages, 10 figure

Nonlinear dynamics of coupled transverse-rotational waves in granular chains

The nonlinear dynamics of coupled waves in one-dimensional granular chains with and without a substrate is theoretically studied accounting for quadratic nonlinearity. The multiple time scale method is used to derive the nonlinear dispersion relations for infinite granular chains and to obtain the wave solutions for semiinfinite systems. It is shown that the sum-frequency and difference-frequency components of the coupled transverse-rotational waves are generated due to their nonlinear interactions with the longitudinal wave. Nonlinear resonances are not present in the chain with no substrate where these frequency components have low amplitudes and exhibit beating oscillations. In the chain positioned on a substrate two types of nonlinear resonances are predicted. At resonance, the fundamental frequency wave amplitudes decrease and the generated frequency component amplitudes increase along the chain, accompanied by the oscillations due to the wave numbers asynchronism. The results confirm the possibility of a highly efficient energy transfer between the waves of different frequencies, which could find applications in the design of acoustic devices for energy transfer and energy rectification

KDamping: A Stiffness Based Vibration Absorption Concept

© 2016, © The Author(s) 2016. The KDamper is a novel passive vibration isolation and damping concept, based essentially on the optimal combination of appropriate stiffness elements, which include a negative stiffness element. The KDamper concept does not require any reduction in the overall structural stiffness, thus overcoming the corresponding inherent disadvantage of the “Quazi Zero Stiffness” (QZS) isolators, which require a drastic reduction of the structure load bearing capacity. Compared to the traditional Tuned Mass damper (TMD), the KDamper can achieve better isolation characteristics, without the need of additional heavy masses, as in the case of the T Tuned Mass damper. Contrary to the TMD and its variants, the KDamper substitutes the necessary high inertial forces of the added mass by the stiffness force of the negative stiffness element. Among others, this can provide comparative advantages in the very low frequency range. The paper proceeds to a systematic analytical approach for the optimal design and selection of the parameters of the KDamper, following exactly the classical approach used for the design of the Tuned Mass damper. It is thus theoretically proven that the KDamper can inherently offer far better isolation and damping properties than the Tuned Mass damper. Moreover, since the isolation and damping properties of the KDamper essentially result from the stiffness elements of the system, further technological advantages can emerge, in terms of weight, complexity and reliability. A simple vertical vibration isolation example is provided, implemented by a set of optimally combined conventional linear springs. The system is designed so that the system presents an adequate static load bearing capacity, whereas the Transfer Function of the system is below unity in the entire frequency range. Further insight is provided to the physical behavior of the system, indicating a proper phase difference between the positive and the negative stiffness elastic forces. This fact ensures that an adequate level of elastic forces exists throughout the entire frequency range, able to counteract the inertial and the external excitation forces, whereas the damping forces and the inertia forces of the additional mass remain minimal in the entire frequency range, including the natural frequencies. It should be mentioned that the approach presented does not simply refer to discrete vibration absorption device, but it consists a general vibration absorption concept, applicable also for the design of advanced materials or complex structures. Such a concept thus presents the potential for numerous implementations in a large variety of technological applications, whereas further potential may emerge in a multi-physics environment.status: publishe