Nonlinear interactions and chaotic dynamics of suspended cables with three-to-one internal resonances

Nonlinear planar oscillations of suspended cables subjected to external excitations with three-to-one internal resonances are investigated. At first, the Galerkin method is used to discretize the governing nonlinear integral–partial-differential equation. Then, the method of multiple scales is applied to obtain the modulation equations in the case of primary resonance. The equilibrium solutions, the periodic solutions and chaotic solutions of the modulation equations are also investigated. The Newton–Raphson method and the pseudo-arclength path-following algorithm are used to obtain the frequency/force–response curves. The supercritical Hopf bifurcations are found in these curves. Choosing these bifurcations as the initial points and applying the shooting method and the pseudo-arclength path-following algorithm, the periodic solution branches are obtained. At the same time, the Floquet theory is used to determine the stability of the periodic solutions. Numerical simulations are used to illustrate the cascades of period-doubling bifurcations leading to chaos. At last, the nonlinear responses of the two-degree-of-freedom model are investigated.     

Nonlinear interactions in the planar dynamics of cable-stayed beam

An analytical model is proposed to study the nonlinear interactions between beam and cable dynamics in stayed-systems. The integro-differential problem, describing the in-plane motion of a simple cable-stayed beam, presents quadratic and cubic nonlinearities both in the cable equation and at the boundary conditions. Mainly studied are the effects of quadratic interactions, appearing at relatively low oscillation amplitude. To this end an analysis of the sensitivity of modal properties to parameter variations, in intervals of technical interest, has evidenced the occurrence of one-to-two and two-to-one internal resonances between global and local modes. The interactions between the resonant modes evidences two different sources of oscillation in cables, illustrated by simple 2dof discrete models.In the one-to-two global–local resonance, a novel mechanism is analyzed, by which cable undergoes large periodic and chaotic oscillations due to an energy transfer from the low-global to high-local frequencies.In two-to-one global–local resonance, the well-known parametric-induced cable oscillation in stayed-systems is correctly reinterpreted through the autoparametric resonance between a global and a local mode. Increasing the load the saturation of the global oscillations evidences the energy transfer from high-global to low-local frequencies, producing large cable oscillations. In both cases, the effects of detuning from internal and external resonance are presented.     

Nonlinear dynamics of micromechanical resonator arrays for mass sensing

Nonlinear Dynamics - This paper investigates the mass sensing capability of an array of a few identical electrostatically actuated microbeams, as a first step toward the implementation of arrays of...     

Nonlinear vibrations of a beam with time-varying rigidity and mass

Nonlinear Dynamics - We consider asymptotic solutions for nonlinear beams that can be described by a fourth order hyperbolic equation with an integral nonlinearity and some space and time dependent...     

Nonlinear dynamics of a spinning shaft with non-constant rotating speed

Research on spinning shafts is mostly restricted to cases of constant rotating speed without examining the dynamics during their spin-up or spin-down operation. In this article, initially the equations of motion for a spinning shaft with non-constant speed are derived, then the system is discretised, and finally a nonlinear dynamic analysis is performed using multiple scales perturbation method. The system in first-order approximation takes the form of two coupled sets of paired equations. The first pair describes the torsional and the rigid body rotation, whilst the second consists of the equations describing the two lateral bending motions. Notably, equations of the lateral bending motions of first-order approximation coincide with the system in case of constant rotating speed, and considering the amplitude modulation equations, as it is shown, there are detuning frequencies from the Campbell diagram. The nonlinear normal modes of the system have been determined analytically up to the second-order approximation. The comparison of the analytical solutions with direct numerical simulations shows good agreement up to the validity of the performed analysis. Finally, it is shown that the Campbell diagram in the case of spin-up or spin-down operation cannot describe the critical situations of the shaft. This work paves the way, for new safe operational ‘modes’ of rotating structures bypassing critical situations, and also it is essential to identify the validity of the tools for defining critical situations in rotating structures with non-constant rotating speeds, which can be applied not only in spinning shafts but in all rotating structures.     

Nonlinear dynamics of a satellite with deployable solar panel arrays

The multibody dynamics of a satellite in circular orbit, modeled as a central body with two hinge-connected deployable solar panel arrays, is investigated. Typically, the solar panel arrays are deployed in orbit using preloaded torsional springs at the hinges in a near symmetrical accordion manner, to minimize the shock loads at the hinges. There are five degrees of freedom of the interconnected rigid bodies, composed of coupled attitude motions (pitch, yaw and roll) of the central body plus relative rotations of the solar panel arrays. The dynamical equations of motion of the satellite system are derived using Kane's equations. These are then used to investigate the dynamic behavior of the system during solar panel deployment via the 7-8th-order Runge-Kutta integration algorithms and results are compared with approximate analytical solutions. Chaotic attitude motions of the completely deployed satellite in circular orbit under the influence of the gravity-gradient torques are subsequently investigated analytically using Melnikov's method and confirmed via numerical integration. The Hamiltonian equations in terms of Deprit's variables are used to facilitate the analysis.     

Nonlinear vibration and thermal-buckling of a heated annular plate with a rigid mass

On the basis of Hamilton's principle and dynamic version of vonKàrmàn's equations,the nonlinear vibration and thermal-buckling of a uniformly heated isotropic annular plate with a completely clamped outer edge and a fixed rigid mass along the inner edge are studied. By parametric perturbation and numerical differentiation, the nonlinear response of the plate-mass system and the critical temperature in the mid-plane at which the plate is in buckled state are obtained. Some meaningful characteristic curves and data tables are given.     

Nonlinear dynamics of a Jeffcott rotor with torsional deformations and rotor-stator contact

The dynamics of a modified Jeffcott rotor is studied, including rotor torsional deformation and rotor-stator contact. Conditions are studied under which the rotor undergoes either forward synchronous whirling or self-excited backward whirling motions with continuous stator contact. For forward whirling, the effect on the response is investigated for two commonly used rotor-stator friction models, namely, the simple Coulomb friction and a generalized Coulomb law with cubic dependence on the relative slip velocity. For cases with and without the rotor torsional degree of freedom, analytical estimates and numerical bifurcation analyses are used to map out regions in the space of drive speed and a friction parameter, where rotor-stator contact exists. The nature of the bifurcations in which stability is lost are highlighted. For forward synchronous whirling fold, Hopf, lift-off, and period-doubling bifurcations are encountered. Additionally, for backward whirling, regions of transitions from pure sticking to stick-slip oscillations are numerically delineated.     

Nonlinear dynamics of a resonant gas sensor

We develop a mathematical model for a resonant gas sensor made up of an microplate electrostatically actuated and attached to the end of a cantilever microbeam. The model considers the microbeam as a continuous medium, the plate as a rigid body, and the electrostatic force as a nonlinear function of the displacement and the voltage applied underneath the microplate. We derive closed-form solutions to the static and eigenvalue problems associated with the microsystem. The Galerkin method is used to discretize the distributed-parameter model and, thus, approximate it by a set of nonlinear ordinary-differential equations that describe the microsystem dynamics. By comparing the exact solution to that associated with the reduced-order model, we show that using the first mode shape alone is sufficient to approximate the static behavior. We employ the Finite Difference Method (FDM) to discretize the orbits of motion and solve the resulting nonlinear algebraic equations for the limit cycles. The stability of these cycles is determined by combining the FDM discretization with Floquet theory. We investigate the basin of attraction of bounded motion for two cases: unforced and damped, and forced and damped systems. In order to detect the lower limit of the forcing at which homoclinic points appear, we conduct a Melnikov analysis. We show the presence of a homoclinic point for a loading case and hence entanglement of the stable and unstable manifolds and non-smoothness of the boundary of the basin of attraction of bounded motion.     

Nonlinear dynamics of a regenerative cutting process

We examine the regenerative cutting process by using a single degree of freedom nonsmooth model with a friction component and a time delay term. Instead of the standard Lyapunov exponent calculations, we propose a statistical 0-1?test analysis for chaos detection. This approach reveals the nature of the cutting process signaling regular or chaotic dynamics. For the investigated deterministic model, we are able to show a transition from chaotic to regular motion with increasing cutting speed. For two values of time delay showing the different response, the results have been confirmed by the means of the spectral density and the multiscaled entropy.     

Nonlinear dynamics of a new electro-vibro-impact system

A variety of nonlinear dynamic responses for a new electro-vibro-impact system is presented, with indication of chaotic behavior. By mathematical modeling of the physical system, an insight is obtained to the global system dynamics. The modeling has established a good correlation with experimental data, and hence can be used as a numerical tool to optimize the system dynamics. In particular, with respect to impact forces and progression rate, may then be achieved with minimal computational cost.     

Bubble dynamics and interactions with a pair of micro pillars in tandem

This study investigates flow patterns and bubble dynamics of two-phase flow around two 100 μm diameter circular pillars in tandem, which were entrenched inside a horizontal micro channel. Bubble velocity, trajectory, size, and void fraction were measured using a high speed camera and analyzed using a particle tracking velocimetry method. A range of gas and liquid superficial velocities were tested, resulting in different bubbly flow patterns, which were consistent with previous studies. These flow patterns were altered as they interacted with the pillars. Depending on the relative transverse location of bubbles to the pillars, and through bubble–bubble interaction, the flow sometimes returned to its original state. It was also determined that the pillars altered both the bubble trajectory and void fraction, especially in the pillars region.     

Nonlinear oscillations of a varying mass reservoir

Institute of Mechanics, Ukrainian Academy of Sciences, Kiev. Translated from Prikladnaya Mekhanika, Vol. 26, No. 10, pp. 109–117, October, 1990.     

Nonlinear dynamics of a spur gear pair with force-dependent mesh stiffness

Time-varying mesh stiffness is one of the main excitation sources of a gear system, and it is also considered as an important factor for the vibration and noise of gears. Thus, this excitation is usually taken as an input into the gear dynamic model to obtain the system dynamic responses. However, the mesh stiffness of a gear pair is actually nonlinear with respect to the dynamic mesh force (DMF) that fluctuates during the operation of gears. Therefore, the dynamic model of gears with the quasi-static mesh stiffness calculated under a constant load is not accurate sufficiently. In this paper, a dynamic model of spur gear is established with considering the effect of the force-dependent time-varying mesh stiffness, backlash and profile deviation. Due to the nonlinear relationship between the mesh stiffness and the load for each tooth pair, it needs first to determine the load sharing among tooth pairs and then calculate the overall mesh stiffness of the gear pair. As the mesh stiffness and DMF are related, the mesh stiffness is no longer directly taken into the gear dynamic model as an input, but is jointly solved with the numerical integration process using the gear dynamic model. Finally, the dynamic responses predicted from the established gear dynamic model are compared with the experimental results for validation and compared with the traditional models to reveal their differences. The results indicate that the established dynamic model of spur gear transmission has a wider application range than the traditional models.     

Nonlinear stochastic dynamics of a rub-impact rotor system with probabilistic uncertainties

Nonlinear Dynamics - This paper presents a stochastic model for performing the uncertainty and sensitivity analysis of a Jeffcott rotor system with fixed-point rub-impact and multiple uncertain...     

Nonlinear interactions in a parametrically excited system with widely spaced frequencies

An investigation is presented into the transfer of energy from high- to low-frequency modes. The method of averaging is used to analyze the response of a two-degree-of-freedom system with widely spaced frequencies and cubic nonlinearities to a principal parametric resonance of the high-frequency mode. The conditions under which energy can be transferred from high- to low-frequency modes, as observed in the experiments, are determined. The interactions between the widely separated modes result in various bifurcations, the coexistence of multiple attractors, and chaotic attractors. The results show that damping may be destabilizing. The analytical results are validated by numerically solving the original system.     

Nonlinear dynamics of a viscoelastic sandwich beam with parametric excitations and internal resonance

Nonlinear dynamical behaviors of an axially accelerating viscoelastic sandwich beam subjected to three-to-one internal resonance and parametric excitations resulting from simultaneous velocity and tension fluctuations are investigated. The direct method of multiple scales is adopted to obtain a set of first-order ordinary differential equations and associated boundary conditions. The frequency and amplitude response curves along with their stability and bifurcation are numerically studied. A great number of dynamic behaviors are presented in the form of phase portraits, time traces, Poincaré sections, and FFT power spectra. Due to modal interaction, various periodic, quasiperiodic, and chaotic behaviors are displayed, depending on the initial conditions. The largest Lyapunov exponent is carried out to determine the midly chaotic response by the convergent form of exponents. Numerical results show various oscillatory behaviors indicating the influence of internal resonance and coupled effects of fluctuating axial velocity and tension.