Invariant submodels of the Westervelt model with dissipation

We study three-dimensional Westervelt model of nonlinear hydroacoustics with dissipation. We received all its invariant submodels. With the help of invariant solutions, we explored some wave processes, specifying their physical meaning. The boundary value problems describing these processes are reduced to the nonlinear integro-differential equations. We established the existence and uniqueness of the solutions of these boundary value problems under some additional conditions. Also we considered the invariant solutions of rank 2 and 3. Mechanical relevance of the obtained solutions is as follows: (1) these solutions describe nonlinear and diffraction effects in ultrasonic fields of a special kind, (2) these solutions can be used as a test solutions in the numerical calculations performed in studies of ultrasonic fields generated by powerful emitters.     

Invariant submodels and exact solutions of Khokhlov–Zabolotskaya–Kuznetsov model of nonlinear hydroacoustics with dissipation

We study three-dimensional Khokhlov–Zabolotskaya–Kuznetsov (KZK) model of the nonlinear hydroacoustics with dissipation. This model is described by third order quasilinear partial differential equation of the (KZK). We obtained that the (KZK) equation admits an infinite Lie group of the transformations, depending on the three arbitrary functions. This is due to the fact that in the (KZK) model the main direction of the wave’s propagation is singled out. The submodels of the (KZK) model.are described by the invariant solutions of the (KZK) equation. We studied essentially distinct, not linked by means of the point transformations, invariant solutions of rank 0 and 1 of this equation. Also we considered the invariant solutions of rank 2 and 3. The invariant solutions of rank 0 and 1 are found either explicitly, or their search is reduced to the solution of the nonlinear integro-differential equations. For example, we obtained the invariant solutions that we called by “Ultrasonic knife” and “Ultrasonic destroyer”. The submodel “Ultrasonic knife” have the following property: at each fixed moment of the time in the field of the existence of the solution near a some plane the pressure increases indefinitely and becomes infinite on this plane. The submodel “Ultrasonic destroyer” contains a countable number of “Ultrasonic knives”. The presence of the arbitrary constants in the integro-differential equations, that determine invariant solutions of rank 1 provides a new opportunities for analytical and numerical study of the boundary value problems for the received submodels, and, thus, for the original (KZK) model. With a help of these invariant solutions we researched a propagation of the intensive acoustic waves (one-dimensional, axisymmetric and planar) for which the acoustic pressure, speed and acceleration of its change, or the acoustic pressure , speed and acceleration of its change in the radial direction, or the acoustic pressure, speed and acceleration of its change in the direction of one of the axes are specified at the initial moment of the time at a fixed point. Under the certain additional conditions, we established the existence and the uniqueness of the solutions of boundary value problems, describing these wave processes. Mechanical relevance of the obtained solutions is as follows: (1) these solutions describe nonlinear and diffraction effects in ultrasonic fields of a special kind, (2) these solutions can be used as a test solutions in the numerical calculations performed in studies of ultrasonic fields generated by powerful emitters. Application of the obtained formula generating the new solutions for the found solutions gives families of the solutions containing three arbitrary functions.     

Submodels of model of nonlinear diffusion with non-stationary absorption

We study the model, describing a nonlinear diffusion process (or a heat propagation process) in an inhomogeneous medium with non-stationary absorption (or source). We found tree submodels of the original model of the nonlinear diffusion process (or the heat propagation process), having different symmetry properties. We found all invariant submodels. All essentially distinct invariant solutions describing these invariant submodels are found either explicitly, or their search is reduced to the solution of the nonlinear integral equations. For example, we obtained the invariant solution describing the nonlinear diffusion process (or the heat distribution process) with two fixed "black holes", and the invariant solution describing the nonlinear diffusion process (or the heat distribution process) with the fixed "black hole" and the moving "black hole". The presence of the arbitrary constants in the integral equations, that determine these solutions provides a new opportunities for analytical and numerical study of the boundary value problems for the received submodels, and, thus, for the original model of the nonlinear diffusion process (or the heat distribution process). For the received invariant submodels we are studied diffusion processes (or heat distribution process) for which at the initial moment of the time at a fixed point are specified or a concentration (a temperature) and its gradient, or a concentration (a temperature) and its rate of change. Solving of boundary value problems describing these processes are reduced to the solving of nonlinear integral equations. We are established the existence and uniqueness of solutions of these boundary value problems under some additional conditions. The obtained results can be used to study the diffusion of substances, diffusion of conduction electrons and other particles, diffusion of physical fields, propagation of heat in inhomogeneous medium.     

Invariant solutions in a channel flow using a minimal restricted nonlinear model

Simulations using a Restricted Nonlinear (RNL) system, where mean flow distortion resulting from Reynolds stress feedback regenerates rolls, is applied in a channel flow under subcritical conditions. This quasi-linear restriction of the dynamics is used to study invariant solutions located in the bulk of the flow found recently by Rawat et al. (2016) [14]. It is shown that the RNL system truncated to a single streamwise mode for the perturbation supports invariant solutions that are found to bifurcate from a relative periodic orbit into a travelling wave solution when the spanwise size is increasing. In particular, the travelling wave solution exhibits a spanwise localized structure that remains unchanged for large values of the spanwise extent as the invariant solution lying on the lower branch found by Rawat et al. (2016) [14]. In addition, travelling wave solutions provided by this minimal RNL system are self-similar with respect to the Reynolds number based on the centreline velocity, and the half-channel height varying from 2000 to 5000.     

Invariant submodels of the model of thermal motion of gas in a rarefied space

A model describing the thermal motion of a gas in a rarefied space is investigated. This model can be used in the study of the motion of gas in outer space, and the processes occurring inside the tornado, and the state of the medium behind the shock front of the wave after a very intense explosion. For a given initial pressure distribution, a special choice of mass Lagrange variables leads to a reduced system of differential equations describing this motion, in which the number of independent variables is one less than the original system. This means that there is a stratification of a highly rarefied gas with respect to pressure. Namely, in a strongly rarefied space for each given initial pressure distribution, at each instant of time all gas particles are localized on a two-dimensional surface moving in this space. At each point of this surface, the acceleration vector is collinear with its normal vector. The resulting system admits an infinite Lie transformation group. All significantly various submodels that are invariant with respect to the subgroups of its eight-parameter subgroup generated by the transfer, extension, rotation, and hyperbolic rotation operators (the Lorentz operator) are found. For invariant submodels of rank 1, the basic mechanical characteristics of the gas flow described by them are obtained. Conditions for the existence of these submodels are given. For invariant submodels of rank 2, integral equations describing these submodels are obtained. For some submodels, the problem of describing the gas flow from the initial location of its particles and the distribution of their velocities has been investigated.     

A numerical model for the nonlinear interaction of elastic waves with cracks

Microstructures such as cracks and microfractures play a significant role in the nonlinear interactions of elastic waves, but the precise mechanism of why and how this works is less clear. Here we simulate wave propagation to understand these mechanisms, complementing existing theoretical and experimental works.We implement two models, one of homogeneous nonlinear elasticity and one of perturbations to cracks, and then use these models to improve our understanding of the relative importance of cracks and intrinsic nonlinearity. We find, by modeling the perturbations in the speed of a low-amplitude P-wave caused by the propagation of a large-amplitude S-wave that the nonlinear interactions of P- and S-waves with cracks are significant when the particle motion is aligned with the normal to the crack face, resulting in a larger magnitude crack dilation. This improves our understanding of the relationship between microstructure orientations and nonlinear wave interactions to allow for better characterization of fractures for analyzing processes including earthquake response, reservoir properties, and non-destructive testing.     

Dynamic response of a deepwater riser subjected to combined axial and transverse excitation by the nonlinear coupled model

In offshore engineering long slender risers are simultaneously subjected to both axial and transverse excitations. The axial load is the fluctuating top tension which is induced by the floater’s heave motion, while the transverse excitation comes from environmental loads such as waves. As the time-varying axial load may trigger classical parametric resonance, dynamic analysis of a deepwater riser with combined axial and transverse excitations becomes more complex. In this study, to fully capture the coupling effect between the planar axial and transverse vibrations, the nonlinear coupled equations of a riser’s dynamic motion are formulated and then solved by the central difference method in the time domain. For comparison, numerical simulations are carried out for both linear and nonlinear models. The results show that the transverse displacements predicted by both models are similar to each other when only the random transverse excitation is applied. However, when the combined axial dynamic tension and transverse wave forces are both considered, the linear model underestimates the response because it ignores the coupling effect. Thus the coupled model is more appropriate for deep water. It is also found that the axial excitation can significantly increase the riser’s transverse response and hence the bending stress, especially for cases when the time-varying tension is located at the classical parametric resonance region. Such time-varying effects should be taken into account in fatigue safety assessment.     

Invariant formulation of the electromechanical enthalpy function of transversely isotropic piezoelectric materials

Summary Theoretical and numerical aspects of the formulation of electromechanically coupled, transversely isotropic solids are discussed within the framework of the invariant theory. The main goal is the representation of the governing constitutive equations for reversible material behaviour based on an anisotropic electromechanical enthalpy function, which automatically fulfills the requirements of material symmetry. The introduction of a preferred direction in the argument list of the enthalpy function allows the construction of isotropic tensor functions, which reflect the inherent geometrical and physical symmetries of the polarized medium. After presenting the general framework, we consider two important model problems within this setting: i) the linear piezoelectric solid; and ii) the nonlinear electrostriction. A parameter identification of the invariant- and the common coordinate-dependent formulation is performed for both cases. The tensor generators for the stresses, electric displacements and the moduli are derived in detail, and some representative numerical examples are presented.We thank Dipl.-Ing. H. Romanowski for his support and helpful remarks.     

On the nonlinear dynamics of the traveling-wave solutions of the Serre system

We numerically study nonlinear phenomena related to the dynamics of traveling wave solutions of the Serre equations including the stability, the persistence, the interactions and the breaking of solitary waves. The numerical method utilizes a high-order finite-element method with smooth, periodic splines in space and explicit Runge–Kutta methods in time. Other forms of solutions such as cnoidal waves and dispersive shock waves are also considered. The differences between solutions of the Serre equations and the Euler equations are also studied.     

A two-way model for nonlinear acoustic waves in a non-uniform lattice of Helmholtz resonators

Propagation of high amplitude acoustic pulses is studied in a 1D waveguide connected to a lattice of Helmholtz resonators. An homogenized model has been proposed by Sugimoto (1992), taking into account both the nonlinear wave propagation and various mechanisms of dissipation. This model is extended here to take into account two important features: resonators of different strengths and back-scattering effects. An energy balance is obtained, and a numerical method is developed. A closer agreement is reached between numerical and experimental results. Numerical experiments are also proposed to highlight the effect of defects and of disorder.     

On the nonlinear interactions and existence of breathers in a chain of coupled pendulums

The paper considers a chain of linearly coupled pendulums. Continues first order system equations are treated via time and space multiple scale method which lead to nonlinear Schrödinger equation. Further investigations on the nonlinear Schrödinger equation detects systems responses in the form of propagated nonlinear waves as functions of their envelope and phases. This provides information about localization of nonlinear waves and their directions in space and time.     

Orbital stability of periodic waves for the nonlinear Schrödinger equation

The nonlinear Schr?dinger equation has several families of quasi-periodic traveling waves, each of which can be parametrized up to symmetries by two real numbers: the period of the modulus of the wave profile, and the variation of its phase over a period (Floquet exponent). In the defocusing case, we show that these travelling waves are orbitally stable within the class of solutions having the same period and the same Floquet exponent. This generalizes a previous work (Gallay and Haragus, J. Diff. Equations, 2007) where only small amplitude solutions were considered. A similar result is obtained in the focusing case, under a non-degeneracy condition which can be checked numerically. The proof relies on the general approach to orbital stability as developed by Grillakis, Shatah, and Strauss, and requires a detailed analysis of the Hamiltonian system satisfied by the wave profile.     

Experimental study and electromechanical model analysis of the nonlinear deformation behavior of IPMC actuators

This paper develops analytical electromechanical formulas to predict the mechanical deformation of ionic polymer–metal composite(IPMC) cantilever actuators under DC excitation voltages. In this research, IPMC samples with Pt and Ag electrodes were manufactured, and the large nonlinear deformation and the effect of curvature on surface electrode resistance of the IPMC samples were investigated experimentally and theoretically. A distributed electrical model was modified for calculating the distribution of voltage along the bending actuator. Then an irreversible thermodynamic model that could predict the curvature of a unit part of an IPMC actuator is combined with the electrical model so that an analytical electromechanical model is developed. The electromechanical model is then validated against the experimental results obtained from Pt-and Ag-IPMC actuators under various excitation voltages. The good agreement between the electromechanical model and the actuators shows that the analytical electromechanical model can accurately describe the large nonlinear quasi-static deflection behavior of IPMC actuators.     

Response regimes in equivalent mechanical model of strongly nonlinear liquid sloshing

Equivalent mechanical model of liquid sloshing in partially-filled cylindrical vessel is treated in the cases of free oscillations and of horizontal base excitation. The model is designed to cover both regimes of linear and essentially nonlinear sloshing. The latter regime involves hydraulic impacts applied to the walls of the vessel. These hydraulic impacts are commonly simulated with the help of high-power potential and dissipation functions. For analytical treatment, we substitute this traditional approach by consideration of the impacts with velocity-dependent restitution coefficient. The resulting model is similar to recently explored vibro-impact nonlinear energy sink (VI NES) attached to externally forced linear oscillator. This similarity allowed exploration of possible response regimes. Steady-state and chaotic strongly modulated responses are encountered. Besides, we simulated the responses to horizontal excitation with addition of Gaussian white noise, and related them to reduced dynamics of the system on a slow invariant manifold (SIM). All analytical predictions are in good agreement with direct numerical simulations of the initial reduced-order model.     

A generalized mathematical model to analyze the nonlinear behavior of a controlled gyroscope in gimbals

In this paper, the effect of the parameter variation on the stability and dynamic behavior of a gyroscope in gimbals with a feedback control system, formed by a Proportional + Integral $+$ Derivative (PID) controller and a DC motor with an ideal train gear is researched. The generalized mathematical model of the gyro is obtained from the Euler-Lagrange equations by using the nutation theory of the gyroscope. The use of approximated models of the control system are deduced from the mathematical model of the gyro, taking into account that the integral action of the PID controller is constrained and that the inductance of the DC motor may be negligible. The analysis and choice of appropriate state variables to simulate the dynamic behavior of different models of the gyro are also considered. The paper shows that from the analysis of the equilibrium points, a Bogdanov Takens and a Poincaré-Andronov-Hopf bifurcation can appear. These bifurcations are analyzed from the calculation of a parameter which is known as the first Lyapunov value, showing that it is possible to deduce a procedure to find out when a complicated model can be substituted by a simpler one. In particular, the possibility of self-oscillating and chaotic behavior for different models of the system by using the first Lyapunov value as a function of the parameters of the PID controller is researched. Numerical simulations have been performed to evaluate the analytical results and the mathematical discussions.     

Viscoplastic behaviour of a medium density polyethylene (MDPE): Constitutive equations based on double nonlinear deformation model

The viscoplastic behaviour of a medium density ethylene–butene copolymer (MDPE) is investigated by using samples cut out from thick-walled MDPE pipe. Extensive experimental work has been performed to characterise the nonlinear time-dependent response of such semi-crystalline thermoplastic material. Tests were carried out at 60 °C, on smooth tensile, full axisymmetrically notched creep tensile (FNCT) and double edge notched tensile (DENT) specimens.     

Dynamical behavior of traveling wave solutions of ion acoustic plasma equations

By using the theory of planar dynamical systems to the ion acoustic plasma equations, we obtain the existence of the solutions of the smooth and non-smooth solitary waves and the uncountably infinite smooth and non-smooth periodic waves. Under the given parametric conditions, we present the sufficient conditions to guarantee the existence of the above solutions.     

On the response of rubbers at high strain rates—I. Simple waves

In this series of papers, we examine the propagation of waves of finite deformation in rubbers through experiments and analysis; in the present paper, Part I, attention is focused on the propagation of one-dimensional waves in strips of natural, latex and synthetic, nitrile rubber. Tensile wave propagation experiments were conducted at high strain rates by holding one end fixed and displacing the other end at a constant velocity. A high-speed video camera was used to monitor the motion and to determine the evolution of strain and particle velocity in rubber strips. Analysis of the response through the theory of finite waves indicated a need for an appropriate constitutive model for rubber; by quantitative matching between the experimental observations and analytical predictions, an appropriate instantaneous elastic response for the rubbers was obtained. This matching process suggested that a simple power-law constitutive model was capable of representing the high strain-rate response for both rubbers used.     

Equivalent nonlinear beam model for the 3-D analysis of shear-type buildings: Application to aeroelastic instability

Tower buildings can be very sensitive to dynamic actions and their dynamic analysis is usually carried out numerically through sophisticated finite element models. In this paper, an equivalent nonlinear one-dimensional shear–shear–torsional beam model immersed in a three-dimensional space is introduced to reproduce, in an approximate way, the dynamic behavior of tower buildings. It represents an extension of a linear beam model recently introduced by the authors, accounting for nonlinearities generated by the stretching of the columns. The constitutive law of the beam is identified from a discrete model of a 3D-frame, via a homogenization process, which accounts for the rotation of the floors around the tower axis. The macroscopic shear strain in the equivalent beam is produced by the bending of columns, accompanied by negligible rotation of the floors. A coupled nonlinear shear–torsional mechanical model is thus obtained. The coupling between shear and torsion is related to a non-symmetric layout of the columns, while mechanical nonlinearities are proportional to the slenderness of the columns. The model can be used for the analysis of the response of tower buildings to any kind of dynamic and static excitation. A first application is here presented to investigate the effect of mechanical and aerodynamic coupling on the critical galloping conditions and on the postcritical behavior of tower buildings, based on a quasi-steady model of aerodynamic forces.     

The role of dust charging frequency in the linear and nonlinear propagation of dust acoustic waves

The effect of dust charge fluctuation on the existence and propagation of dust acoustic waves in unmagnetized dusty plasma is studied. Dispersion relation for the dust acoustic waves with temporal dust charge fluctuations is calculated based on the reductive perturbation technique. By considering two different limiting cases of the dust charging frequency, it is shown that the dusty plasma system gives rise to linear or nonlinear dust acoustic waves. The well-known collisionless damping of two existing normal modes and one purely damped mode are recovered when the dust charging frequency is comparable to the dust acoustic wave frequency. On the other hand, when the dust charging frequency is considerably higher than the wave frequency, it is then possible to derive a nonlinear Schrödinger type equation with envelope soliton as solution.