Active Control of a Rectangular Thin Plate Via Negative Acceleration Feedback

The dynamic behavior of a rectangular thin plate under parametric and external excitations is investigated. The motion of the thin plate is modeled by coupled second-order nonlinear ordinary differential equations. Their approximate solutions are sought by applying the method of multiple scales. A reduced system of four first-order ordinary differential equations is determined to describe the time variation of the amplitudes and phases of the vibration in the horizontal and vertical directions. The steady-state response and the stability of the solutions for various parameters are studied numerically, using the frequency-response function and the phase-plane methods. It is also shown that the system parameters have different effects on the nonlinear response of the thin plate. Moreover, the chaotic motion of the thin plate is found by numerical simulation.

Correction for ‘On the complete integrability and linearization of certain second-order nonlinear ordinary differential equations’ by V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan (Proc. R. Soc. A 461, 2451–2476. (doi: 10.1098/rspa.2005.1465)). The sentence preceeding equation (4.48) is incorrect and should read as follows.

The objective of this work is to analytically study the nonlinear dynamics of beam flexures with a tip mass undergoing large deflections. Hamilton's principle is utilized to derive the equations governing the nonlinear vibrations of the cantilever beam and the associated boundary conditions. Then, using a single mode approximation, these nonlinear partial differential equations are reduced to two coupled nonlinear ordinary differential equations. These equations are solved analytically using the multiple time scales perturbation technique. Parametric analytical expressions are presented for the time domain response of the beam around and far from its internal resonance state. These analytical results are compared with numerical ones to validate the accuracy of the proposed analytical model. Compared with numerical solution methods, the proposed analytical technique shortens the computational time, offers design insights, and provides a broader framework for modeling more complex flexure mechanisms. The qualitative and quantitative knowledge resulting from this effort is expected to enable the analysis, optimization, and synthesis of flexure mechanisms for improved dynamic performance.

In this paper, the nonlinear vibrational behavior of a sandwich plate with embedded viscoelastic material is studied through the use of constitutive equations with fractional derivatives. The studied sandwich structure is consisted of a viscoelastic core that is located between the faces of functionally graded magneto-electro-elastic (FG-MEE). In order to determine the frequency-dependent feature of the viscoelastic layer, four-parameter fractional derivative model is utilized. The material properties of FG-MEE face sheets have been distributed considering the power law scheme along the thickness. In addition, for derivation of the governing equations on the sandwich plate, first-order shear deformation plate theory along with von Karman-type of kinematic nonlinearity are implemented. The derived partial differential equations (PDEs) have been transformed to the ordinary differential equations (ODEs) through the Galerkin method. After that, the nonlinear vibration equations for the sandwich plate have been solved by multiple time scale perturbation technique. Moreover, for evaluating the effect of different parameters such as electric and magnetic fields, fractional order, the ratio of the core-to-face thickness and the power low index on the nonlinear vibration characteristics of sandwich plates with FG-MEE face sheets, the parametric analysis has been performed. The obtained results revealed the enhanced nonlinear natural frequency through an increment in the fractional order. Furthermore, the prominent influence of fractional order on the nonlinear frequency of sandwich plate was declared at the negative electric potential and positive magnetic potential.

In this paper, the linear absorber is proposed to reduce the vibration of a nonlinear dynamical system at simultaneous primary resonance and the presence of 1:1 internal resonance. This leads to a two-degree-of-freedom system subjected to external excitation force. The method of multiple scales perturbation technique is applied throughout to determine the analytical solution up to first-order approximations. The stability of the system near the one of the worst resonance case is studied using the frequency response equations. The effects of the different system and absorber parameters on the behavior of the main system are studied numerically. For validity, the numerical solution is compared with the analytical solution and gets a good agreement. Effectiveness of the absorber ( $$E_{a})$$ is about 800 for the nonlinear vibrating system. The simulation results are achieved using MATLAB programs. At the end of the work, the comparison with the available published work is reported.

We study the recurrence coefficients of the monic polynomials [Formula: see text] orthogonal with respect to the deformed (also called semi-classical) Freud weight [Formula: see text] with parameters [Formula: see text]. We show that the recurrence coefficients [Formula: see text] satisfy the first discrete Painlevé equation (denoted by d[Formula: see text]), a differential–difference equation and a second-order nonlinear ordinary differential equation (ODE) in [Formula: see text]. Here [Formula: see text] is the order of the Hankel matrix generated by [Formula: see text]. We describe the asymptotic behavior of the recurrence coefficients in three situations: (i) [Formula: see text], [Formula: see text] finite, (ii) [Formula: see text], [Formula: see text] finite, (iii) [Formula: see text], such that the radio [Formula: see text] is bounded away from [Formula: see text] and closed to [Formula: see text]. We also investigate the existence and uniqueness for the positive solutions of the d[Formula: see text]. Furthermore, we derive, using the ladder operator approach, a second-order linear ODE satisfied by the polynomials [Formula: see text]. It is found as [Formula: see text], the linear ODE turns to be a biconfluent Heun equation. This paper concludes with the study of the Hankel determinant, [Formula: see text], associated with [Formula: see text] when [Formula: see text] tends to infinity.

This article concerns the positive periodic solutions for a system of second-order nonlinear ordinary differential equations, in which the nonlinear term is sublinear in one equation and superlinear in the other equation.By using the fixed point theorem of cone expansion and compression we obtain the existence of positive periodic solutions.

59 - Exact Second Order Equations

Damped and driven oscillators are generally modeled with a nonautonomous second-order nonlinear ordinary differential equation including a sinusoidal driving forcing term, such as the forced Duffing equation and the forced Holmes–Rand equation. These equations have been extensively studied during the last century and the last two decades. In the early 1990s, Abarbanel, Rabinovich and Sushchik proposed replacing the sinusoidal forcing term with a “force controlled by the movements of the oscillator itself”, i.e. by the product of two variables: the first being the solution of the oscillator itself, while the second is the solution of a first-order nonlinear ordinary differential equation. They referred to the resulting autonomous dynamical system of two coupled nonlinear ordinary differential equations as a “controlled nonlinear oscillator”. To that end, they introduced a change of variables and parameters to transform the “controlled nonlinear oscillator” that corresponds to a particular case of the forced Duffing equation into the Lorenz system. The aim of this work is to show that their idea can be further generalized and applied to many other dynamical systems, including the forced Holmes–Rand equation, Chua’s cubic circuit, Chen’s system and the forced Helmholtz oscillator. It is proved that a certain class of three-dimensional dynamical systems can be rewritten into the form of “generalized controlled nonlinear oscillators”, which can then be transformed into various Lorenz-like systems. Such a transformation could be very useful for the study of intermittent chaos.

Low Reynolds number flows in a microscopic and tapered tube with a permeability are theoretically investigated in steady conditions. A momentum equation for the main flow in the tube is derived from a set of incompressible Navier-Stokes equations for an axisymmetric flow. The second-order nonlinear ordinary differential equation on a pressure is derived using a velocity profile of the fourth-order polynomials for the main flow, in which there are nonlinear terms of both orders of Re2 and Re. Solutions obtained are classified into two types; one is a flow with a Reynolds number far less than 161, whilst another is a flow with a Reynolds number larger than the former case but smaller than 248. They are compared with an exact solution of the original nonlinear differential equation. Effects of the permeability of the tube wall and the Reynolds number of the main flow upon the pressure loss, the axial and radial components of the main flow velocity, and the resistance coefficient are clarified. Furthermore, a flow in a permeable and straight tube is investigated as a special case of the permeable and tapered tube, and it is demonstrated that an adverse pressure gradient takes place at a relatively large Reynolds number when the main flow is withdrawn through the permeable wall. As a consequence, the results obtained for the permeable and tapered tube contain flows in both the impermeable tapered tube and the impermeable straight tube, the latter of which is Hagen-Poiseuille flow, as special cases.

Acoustic nonlinearities in a quasi 1-D duct with arbitrary mean properties and mean flow

In the present study, forced vibration of a simply supported embedded curved single-walled carbon nanotube (ECSWCNT) subjected to a moving harmonic load is investigated based on nonlocal Euler-Bernoulli beam theory. By using a single-mode Galerkin approximation method, the nonlinear integral-differential equation governing the motion of the nanotube is converted into a second-order nonlinear ordinary differential equation. The differential equation of the model is solved using Magnus expansion method which is one of the geometric integration methods. In the numerical calculation, the effects of nonlocal parameter, aspect ratio of ECSWCNT, velocity and the elastic medium constant is discussed. The results show that the above mentioned effects play an important role on the dynamic behavior of ECSWCNT.

Many boundary-value problems in nonlinear elastostatics for incompressible isotropic materials have been solved analytically. The situation for compressible materials is quite different. The main difficulty is the absence of the simplified kinematics arising from the incompressibility constraint. Considerable progress in the analytic solution of boundary-value problems for classes of compressible materials has however taken place in recent years and further developments in this direction are considered in this paper. A very special topic in the equilibrium theory for finite deformations of homogeneous isotropic compressible materials is investigated, namely that of spherically symmetric or cylindrically symmetric deformations. Since only axisymmetric deformations are considered, the equilibrium equations reduce to a single second-order nonlinear ordinary differential equation for the radial deformation field. In previous work, two main methods for obtaining analytic solutions for these equations have been developed, the first based on a change of variable that allows one to transform the second-order ODE to a pair of first-order ODEs that may be solved in parametric form. In some cases, the parameter can be eliminated to yield an explicit closed form expression for the deformation. The second method is based on the observation that the second-order ODE is invariant under a stretching transformation of the independent and dependent variables and thus may always be transformed to a first-order ODE. The purpose of this paper is to show how the powerful techniques of Lie group analysis (LGA) for ODEs can be used to unify both the above approaches and how LGA is an invaluable guide in the development of analytic solution methods. It is shown that both of the approaches can be derived from Lie algebra invariance criteria and that the second approach is, in fact, a special case of the first. Results from LGA are then applied to investigate the structure and solution set of the governing ODEs. A systematic procedure for solving the axisymmetric equations of compressible finite elastostatics is thereby proposed. The results are illustrated by application to several specific strain-energy density functions.

A new class of self-consistent, one-dimensionally confined, neutral solutions to the stationary Vlasov and Maxwell equations is presented. Apart from the special cases of conventional sheet pinches, these equilibria all display current shear (which, in analogy with magnetic shear, is defined as rotation of the current vector as the plasma is traversed in the confined direction). The new solutions are obtained numerically, after the relativistic Vlasov-Maxwell problem has been reduced to a single second-order nonlinear ordinary differential equation for the critical component of the self-consistent vector potential. The physical properties of representative equilibria are given in graphical form. Included in addition to thin sheet pinches are broader configurations with a region of nearly uniform plasma number density, bounded by rather sharp plasma/magnetic-field interfaces.

A note on the application of Darboux polynomial method to an nonlinear oscillator equation