A comparison of three perturbation methods for non-linear hyperbolic waves

Simple wave solutions of non-linear hyperbolic equations are studied by using the method of renormalization, the analytic method of characteristics, and the method of multiple scales. It is shown that the results of the method of renormalization depend on whether the potential function or the velocity is normalized. This arbitrariness does not occur when using either the analytic method of characteristics or the method of multiple scales. However, special consideration must be given in determining the potential from the velocity obtained by the analytic method of characteristics. No such consideration is needed when the method of multiple scales is applied. The first term obtained for the potential by the method of multiple scales contains a cumulative term in addition to a non-cumulative term. This first-order term is shown to yield the equal area rule for shock waves, and the slope of an equipotential line is the arithmetic mean of the slope of the characteristic in the unperturbed medium and the slope of the characteristic at the point under consideration.     

Analyses of axisymmetric waves in layered piezoelectric rods and their composites

An exact treatment of the propagation of axisymmetric waves in coaxial anisotropic assembly of piezoelectric rod systems is presented. The rod system consists of an arbitrary number of coaxial layers, each possessing transversely isotropic symmetry properties. The treatment, which is based on the transfer matrix technique, is capable of deriving the dispersion relations for a variety of situations. These include the case of a single rod system that is either embedded in an infinitely extended solid or fluid host or kept free. The procedure is also adapted to derive approximate solutions for the cases of a periodic fiber distribution in a matrix material, which model unidirectional fiber-reinforced composites. The results are numerically illustrated for a widely used piezoelectric-polymer composite. It is seen that piezoelectric coupling can significantly change the morphology of the dispersive behavior of the composite.     

Dynamics of a Cubic Nonlinear Vibration Absorber

We study the dynamics of a nonlinear active vibration absorber. We consider a plant model possessing curvature and inertia nonlinearities and introduce a second-order absorber that is coupled with the plant through user-defined cubic nonlinearities. When the plant is excited at primary resonance and the absorber frequency is approximately equal to the plant natural frequency, we show the existence of a saturation phenomenon. As the forcing amplitude is increased beyond a certain threshold, the response amplitude of the directly excited mode (plant) remains constant, while the response amplitude of the indirectly excited mode (absorber) increases. We obtain an approximate solution to the governing equations using the method of multiple scales and show that the system possesses two possible saturation values. Using numerical techniques, we perform stability analyses and demonstrate that the system exhibits complicated dynamics, such as Hopf bifurcations, intermittency, and chaotic responses.     

Nonlinear Nonplanar Dynamics of Parametrically Excited Cantilever Beams

The nonlinear nonplanar response of cantilever inextensional metallic beams to a principal parametric excitation of two of its flexural modes, one in each plane, is investigated. The lowest torsional frequencies of the beams considered are much larger than the frequencies of the excited modes so that the torsional inertia can be neglected. Using this condition as well as the inextensionality condition, we develop a Lagrangian whose variation leads to two integro-partial-differential equations governing the motions of the beams. The method of time-averaged Lagrangian is used to derive four first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the two interacting modes. These modulation equations exhibit symmetry properties. A pseudo arclength scheme is used to trace the branches of the equilibrium solutions and an investigation of the eigenvalues of the Jacobian matrix is used to assess their stability. The equilibrium solutions experience pitchfork, saddle-node, Hopf, and codimension-2 bifurcations. A detailed bifurcation analysis of the dynamic solutions of the modulation equations is presented. Five branches of dynamic (periodic and chaotic) solutions were found. Two of these branches emerge from two Hopf bifurcations and the other three are isolated. The limit cycles undergo symmetry-breaking, cyclic-fold, and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging and boundary crises.     

Phenomena and modeling of piezoelectric energy harvesting from freely oscillating cylinders

We investigate energy harvesting from vortex-induced vibrations of a freely moving rigid circular cylinder with a piezoelectric transducer attached to its transverse degree of freedom. The power levels that can be generated from these vibrations and variations of these levels with the freestream velocity are determined. A mathematical model that accounts for the coupled lift force, cylinder motion, and harvested voltage is presented. Linear analysis is performed to determine the effect of the electrical load resistance of the transducer on the natural frequency of the cylinder and the onset of synchronization (the shedding frequency is equal to the cylinder oscillating frequency) region. The impact of the nonlinearities on the cylinder response and harvested energy is investigated. The results show that the load resistance shifts the onset of synchronization to higher freestream velocities. For two different system parameters, the results show that the nonlinearities result in a hardening behavior for some values of the load resistance.     

Bifurcations in a forced softening duffing oscillator

The response of a damped Duffing oscillator of the softening type to a harmonic excitation is analyzed in a two-parameter space consisting of the frequency and amplitude of the excitation. An approximate procedure is developed for the generation of the bifurcation diagram in the parameter space of interest. It is a combination of second-order perturbation solutions of the system in the neighborhood of its non-linear resonances and Floquet analysis. The results show that the proposed scheme is capable of predicting symmetry-breaking and period-doubling bifurcations as well as Jumps to either bounded or unbounded motions. The results obtained are validated using analogand digital-computer simulations, which show chaos and unbounded motions, among other behaviors.     

Cargo Pendulation Reduction of Ship-Mounted Cranes

Ship-mounted cranes are used to transfer cargo from large container ships to smaller lighters when deep-water ports are not available. The wave-induced motion of the crane ship can produce large pendulations of the cargo being hoisted and cause operations to be suspended. In this work, we show that it is possible to reduce these pendulations significantly by controlling the slew and luff angles of the boom. Such a control can be achieved with the heavy equipment that is already part of the crane so that retrofitting existing cranes would require a small effort. Moreover, the control is superimposed on the commands of the operator transparently. The successful control strategy is based on delayed feedback of the angles of the cargo-hoisting cable in and out of the plane of the boom and crane tower. Its effectiveness is demonstrated in a fully nonlinear three-dimensional computer simulation and in an experiment with a 1/24th-scale model of a T-ACS (The Auxiliary Crane Ship) crane mounted on a platform moving with three degrees of freedom. The results demonstrate that the pendulations can be significantly reduced, and therefore the range of sea conditions in which cargo-transfer operations can take place can be greatly expanded.     

Bifurcations and chaos in parametrically excited single-degree-of-freedom systems

The behavior of single-degree-of-freedom systems possessing quadratic and cubic nonlinearities subject to parametric excitation is investigated. Both fundamental and principal parametric resonances are considered. A global bifurcation diagram in the excitation amplitude and excitation frequency domain is presented showing different possible stable steady-state solutions (attractors). Fractal basin maps for fundamental and principal parametric resonances when three attractors coexist are presented in color. An enlargement of one region of the map for principal parametric resonance reveals a Cantor-like set of fractal boundaries. For some cases, both periodic and chaotic attractors coexist.     

Analysis of one-to-one autoparametric resonances in cables—Discretization vs. direct treatment

We discuss solution methods for nonlinear vibrations of cables having small initial sag-to-span ratios. One-to-one internal resonances between the in-plane and out-of-plane modes as well as primary resonances of the in-plane mode are considered. Approximate solutions are obtained by two different approaches. In the first approach, the method of multiple scales is applied directly to the governing partial-differential equations and boundary conditions. In the second approach, the equations are first discretized, and then the method of multiple scales is applied to the resulting ordinary-differential equations. It is shown that treatment of the discretized system is inaccurate compared to direct treatment of the partial-differential system. Discrepancies between the two solutions appear even at the first level of approximation. Stability analyses of the amplitude and phase modulation equations for both methods are also performed.     

The response of nonlinear systems to modulated high-frequency input

We study the response of a single-degree-of-freedom system with cubic nonlinearities to an amplitude-modulated excitation whose carrier frequency is much higher than the natural frequency of the system. The only restriction on the amplitude modulation is that it contain frequencies much lower than the carrier frequency of the excitation. We apply the theory to different types of amplitude modulation and find that resonant excitation of the system may occur under some conditions.