Subharmonics analysis of nonlinear flexural vibrations of piezoelectrically actuated microcantilevers

Using the method of multiple scales, an extensive frequency response and subharmonic resonance analysis of the equations of motion governing the nonlinear flexural vibrations of piezoelectrically actuated microcantilevers is performed. Such comprehensive understanding of the nonlinear response and subharmonics analysis of these microcantilevers is, indeed, justified by the applications of piezoelectrically actuated microcantilevers that are increasingly becoming popular in many science and engineering areas including scanning force microscopy, biosensors, and microactuators. Along this line, the method of multiple scales is used to derive the 2× and 3× subharmonic resonances appearing in nonlinear flexural vibrations of a piezoelectrically actuated microcantilever. An experimental examination is performed in order to verify the analytical results. The analytical and experimental results yield the same system response for the fundamental frequency. In addition, the experimental results demonstrate the presence of subharmonic resonances that are supported by numerical simulations of the equations of motion. The experimental mode shapes of these subharmonic frequencies are also measured and compared with fundamental frequency.     

Isogeometric analysis of nonlinear Euler–Bernoulli beam vibrations

In this paper we analyze the vibrations of nonlinear structures by means of the novel approach of isogeometric finite elements. The fundamental idea of isogeometric finite elements is to apply the same functions, namely B-Splines and NURBS (Non-Uniform Rational B-Splines), for describing the geometry and for representing the numerical solution. In case of linear vibrational analysis, this approach has already been shown to possess substantial advantages over classical finite elements, and we extend it here to a nonlinear framework based on the harmonic balance principle. As application, the straight nonlinear Euler–Bernoulli beam is used, and overall, it is demonstrated that isogeometric finite elements with B-Splines in combination with the harmonic balance method are a powerful means for the analysis of nonlinear structural vibrations. In particular, the smoother k-method provides higher accuracy than the p-method for isogeometric nonlinear vibration analysis.     

Three-dimensional nonlinear vibrations of composite beams — I. Equations of motion

Newton's second law is used to develop the nonlinear equations describing the extensional-flexural-flexural-torsional vibrations of slewing or rotating metallic and composite beams. Three consecutive Euler angles are used to relate the deformed and undeformed states. Because the twisting-related Euler angle is not an independent Lagrangian coordinate, twisting curvature is used to define the twist angle, and the resulting equations of motion are symmetric and independent of the rotation sequence of the Euler angles. The equations of motion are valid for extensional, inextensional, uniform and nonuniform, metallic and composite beams. The equations contain structural coupling terms and quadratic and cubic nonlinearities due to curvature and inertia. Some comparisons with other derivations are made, and the characteristics of the modeling are addressed. The second part of the paper will present a nonlinear analysis of a symmetric angle-ply graphite-epoxy beam exhibiting bending-twisting coupling and a two-to-one internal resonance.     

Tracking control of uncertain switched nonlinear cascade systems: a nonlinear H ∞ sliding mode control method

This paper considers the tracking control problem for a class of uncertain switched nonlinear cascade systems via the multiple Lyapunov functions (MLFs) method. Each subsystem under consideration is composed of two cascade-connected parts: the null space dynamics part and the range space dynamics part. The two main robust control strategies, nonlinear H _∞ control (NHC) and the sliding mode control (SMC), are integrated to function in a complementary manner for tracking control tasks. Furthermore, sufficient conditions for the solvability of the tracking control problem of the switched system and design of both switching laws and controllers are presented. Finally, a simulation example is provided to demonstrate the feasibility of the developed method.     

Homogenization of the nonlinear Kelvin–Voigt model of viscoelasticity and of the Prager model of plasticity

Denoting by the stress tensor, by the linearized strain tensor, by A the elasticity tensor, and assuming that is a convex potential, the inclusion accounts for nonlinear viscoelasticity, and encompasses both the linear Kelvin–Voigt model of solid-type viscoelasticity and the Prager model of rigid plasticity with linear kinematic strain-hardening. This relation is assumed to represent the constitutive behavior of a space-distributed system, and is here coupled with the dynamical equation. An initial- and boundary-value problem is formulated, and the existence and uniqueness of the solution are proved via classical techniques based on compactness and monotonicity. A composite material is then considered, in which the function and the tensor A rapidly oscillate in space. A two-scale model is derived via Nguetseng’s notion of two-scale convergence. This provides a detailed account of the mesoscopic state of the system. Any dependence on the fine-scale variable is then eliminated, and the existence of a solution of a new single-scale macroscopic model is proved. The final outcome is at variance with the nonlinear extension of the generalized Kelvin–Voigt model, which is based on an apparently unjustified mean-field-type hypothesis.     

Autoparametric amplification of two nonlinear coupled mass–spring systems

The gearboxes of machines generally operate under a time-varying state rather than under steady-state conditions. However, it is difficult to investigate the nonlinear dynamics of a time-varying gear system. A gear system model of a railway vehicle was proposed in consideration of its time-varying mesh stiffness, nonlinear backlash, transmission error, time-varying external excitation, and rail irregularity. To obtain the nonlinear behaviors of a time-varying stochastic gear system, a quasi-static analysis was performed to observe its doubling-periodic bifurcation, chaotic motion, and transition from a lower to a higher power periodic motion. Based on the energy comparison results, the time-varying stochastic gear system was degraded to a time-varying system to simplify the calculation. Furthermore, the nonlinear response of the time-varying system was computed using the Runge–Kutta method and was compared with the results of a quasi-static analysis that employed a short-time Fourier transform method. The results of the quasi-static analysis were consistent with the results of the time–frequency analysis for the time-varying gear system except for the result at 3180 r/min, which represented a short period wherein the process transitioned to chaos. Hence, the comparison demonstrates the applicability of the quasi-static analysis for the nonlinear behavior analysis of a time-varying stochastic system.     

Magnetic field effects on the nonlinear vibration of a rotor

The nonlinear vibration of a rotor operated in a magnetic field with geometric and inertia nonlinearity is investigated. An asymmetric magnetic flux density is generated,resulting in the production of a load on the rotor since the air-gap distribution between the rotor and the stator is not uniform. This electromagnetic load is a nonlinear function of the distance between the geometric centers of the rotor and the stator. The nonlinear equation of motion is obtained by the inclusion of the nonlinearity in the inertia, the curvature, and the electromagnetic load. After discretization of the governing partial differential equations by the Galerkin method, the multiple-scale perturbation method is used to derive the approximate solutions to the equations. In the numerical results, the effects of the electromagnetic parameter load, the damping coefficient, the amplitude of the initial displacement, the mass moment of inertia, and the rotation speed on the linear and nonlinear backward and forward frequencies are investigated. The results show that the magnetic field has significant effects on the nonlinear frequency of oscillation.     

Research of giant magnetostrictive actuator’s nonlinear dynamic behaviours

The multi-coupled nonlinear factors existing in the giant magnetostrictive actuator (GMA) have a serious impact on its output characteristics. If the structural parameters are not properly designed, it is easy to fall into the nonlinear instability, which has seriously hindered its application in many important fields. The electric–magnetic-machine coupled dynamic mathematical model for GMA is established according to J-A dynamic hysteresis model, ampere circuit law, nonlinear quadratic domain model and structure dynamics equation. Nonlinear dynamic analysis method is applied to study the nonlinear dynamic behaviour of the key structure parameters to reveal their influence on the system stability. The design principle of structural parameters is obtained by studying stability of GMA, which provides theoretical basis and technical support for the structural stability design.     

Experimental classification of nonlinear dynamics in microfluidic bubbles’ flow

A nonlinear analysis method based on the evaluation of d-infinite and largest Lyapunov exponent was used to study the complex dynamics of air bubbles carried by water and flowing in a microfluidic snake channel. A rich variety of nonlinear dynamics and flow patterns was found through the experimental observation of bubbles’ motion. The results and their graphical representation show the capability of the proposed set of dimensionless parameters to classify the nonlinearity of the process showing also its sensitivity to input flow variations.     

Experimental and numerical analysis of nonlinear dynamics of rotor–bearing–seal system

Nonlinear dynamics and stability of the rotor–bearing–seal system are investigated both theoretically and experimentally. An experimental rotor–bearing–seal device is designed and corresponding tests are carried out. The experimental rotor system is simplified as the Jeffcott rotor. The nonlinear oil–film forces are obtained under the short bearing theory and Muszynska nonlinear seal force model is used. Numerical method is utilized to solve the nonlinear governing equations. Bifurcation diagrams, waterfall plots, Poincaré maps, spectrum plots and rotor orbits are drawn to analyze various nonlinear phenomena and system unstable processes. Theoretical results from numerical analysis are in good agreement with results from experiments. Conclusions are drawn and prove that this study will contribute to the further understanding of nonlinear dynamics and stability of the rotor system with the fluid-induced forces from oil–film bearings and the seals.     

Order reduction and nonlinear behaviors of a continuous rotor system

An isotropic flexible shaft, acted by nonlinear fluid-induced forces generated from oil-lubricated journal bearings and hydrodynamic seal, is considered in this paper. Dimension reductions of the rotor system were carried out by both the standard Galerkin method and the nonlinear Galerkin method. Numerical simulations provide bifurcation diagrams, spectrum cascade, orbits of the disk center and Poincaré maps, to demonstrate the dynamical behaviors of the system. The results reveal transitions, or bifurcations, of the rotor whirl from being synchronous to non-synchronous as the unstable speed is exceeded. The non-synchronous oil/seal whirl is a quasi-periodic motion. In the regime of quasi-periodic motion, the “windows” of multi-periodic motion were found. The investigation shows that the nonlinear Galerkin method has an advantage over the standard one with the same order of truncations, because the influences of higher modes are considered by the former.     

Vibration control of nonlinear Absorber–Isolator-Combined structure with time-delayed coupling

A nonlinear combined structure consisted of isolator and absorber with time-delayed coupling active control is proposed in this study, whose vibration suppression effectiveness and control mechanism are investigated. The mathematical model of the combined structure is obtained and stability analysis for different structural parameters and time delay are firstly carried out, which provides a general guideline for the ranges of active control parameters. Then the combined effect of nonlinearity and time delay on vibration suppression and energy transfer is discussed in details based on the analysis of control mechanism by the method of multiple scales. Since the time-delayed nonlinear absorber can induce internal resonance between different modes, the vibration energy at low frequencies can be transferred to high frequency mode and the vibration of the fundamental frequency range is thus suppressed. This paper provides a novel application of internal resonance in vibration suppression of an Absorber–Isolator-Combined structure.     

Stability of exact solutions of the cubic-quintic nonlinear Schrödinger equation with periodic potential

The nonlinear Schr?dinger equation with attractive quintic nonlinearity in periodic potential in 1D, modeling a dilute-gas Bose–Einstein condensate in a lattice potential, is considered and one family of exact stationary solutions is discussed. Some of these solutions have an analog neither in the linear Schr?dinger equation nor in the integrable nonlinear Schr?dinger equation. Their stability is examined analytically and numerically.     

The use of Volterra series in the analysis of the nonlinear Schrödinger equation

A Volterra series analysis is used to analyse the dispersive behaviour in the frequency domain for the non-linear Schrödinger equation (NLS). It is shown that the solution of the initial value problem for the nonlinear Schrödinger equation admits a local multi-input Volterra series representation. Higher order spatial frequency responses of the nonlinear Schrödinger equation can therefore be defined in a similar manner as for lumped parameter non-linear systems. A systematic procedure is presented to calculate these higher order spatial frequency response functions analytically. The frequency domain behaviour of the equation, subject to Gaussian initial waves, is then investigated to reveal a variety of non-linear phenomena such as self-phase modulation (SPM), cross-phase modulation (CPM), and Raman effects modelled using the NLS.     

Order reduction of retarded nonlinear systems – the method of multiple scales versus center-manifold reduction

We compare two approaches for determining the normal forms of Hopf bifurcations in retarded nonlinear dynamical systems; namely, the method of multiple scales and a combination of the method of normal forms and the center-manifold theorem. To describe and compare the methods without getting involved in the algebra, we consider three examples: a scalar equation, a single-degree-of-freedom system, and a three-neuron model. The method of multiple scales is directly applied to the retarded differential equations. In contrast, in the second approach, one needs to represent the retarded equations as operator differential equations, decompose the solution space of their linearized form into stable and center subspaces, determine the adjoint of the operator equations, calculate the center manifold, carry out details of the projection using the adjoint of the center subspace, and finally calculate the normal form on the center manifold. We refer to the second approach as center-manifold reduction. Finally, we consider a problem in which the retarded term appears as an acceleration and treat it using the method of multiple scales only. Communicated by G. Rega     

Fractional nonlinear energy sinks

The cubic or third-power(TP) nonlinear energy sink(NES) has been proven to be an effective method for vibration suppression, owing to the occurrence of targeted energy transfer(TET). However, TET is unable to be triggered by the low initial energy input, and thus the TP NES would get failed under low-amplitude vibration. To resolve this issue, a new type of NES with fractional nonlinearity, e.g., one-third-power(OTP)nonlinearity, is proposed. The dynamic behaviors of a linear oscillator(LO) with...     

Dynamics of superregular breathers in the quintic nonlinear Schrödinger equation

In this paper, we consider an extended nonlinear Schrödinger equation that includes fifth-order dispersion with matching higher-order nonlinear terms. Via the modified Darboux transformation and Joukowsky transform, we present the superregular breather (SRB), multipeak soliton and hybrid solutions. The latter two modes appear as a result of the higher-order effects and are converted from a SRB one, which cannot exist for the standard NLS equation. These solutions reduce to a small localized perturbation of the background at time zero, which is different from the previous analytical solutions. The corresponding state transition conditions are given analytically. The relationship between modulation instability and state transition is unveiled. Our results will enrich the dynamics of nonlinear waves in a higher-order wave system.     

Painlevé analysis and invariant solutions of generalized fifth-order nonlinear integrable equation

In present work, new form of generalized fifth-order nonlinear integrable equation has been investigated by locating movable critical points with aid of Painlevé analysis and it has been found that this equation passes Painlevé test for \(\alpha =\beta \) which implies affirmation toward the complete integrability. Lie symmetry analysis is implemented to obtain the infinitesimals of the group of transformations of underlying equation, which has been further pre-owned to furnish reduced ordinary differential equations. These are then used to establish new abundant exact group-invariant solutions involving various arbitrary constants in a uniform manner.     

Multidimensional modal analysis of liquid nonlinear sloshing in right circular cylindrical tank

The multidimensional modal theory proposed by Faltinsen,et al.(2000) is applied to solve liquid nonlinear free sloshing in right circular cylindrical tank for the first time. After selecting the leading modes and fixing the order of magnitudes baaed on the Narimanov-Moiseev third order asymptotic hypothesis,the general infinite dimensional modal system is reduced to a five dimensional asymptotic modal system (the system of second order nonlinear ordinary differential equations coupling the generalized time dependent coordinates of free surface wave elevation).The numerical integrations of this modal system discover most important nonlinear phenomena,which agree well with both pervious analytic theories and experimental observations. The results indicate that the multidimensional modal method is a very good tool for solving liquid nonlinear sloshing dynamics and will be developed to investigate more complex sloshing problem in our following work.