Nonsymmetric Nonlinearity Research Articles

Structural Identifiability Evaluation of System with Nonsymmetric Nonlinearities

The complexity of objects and control systems increases the requirements for mathematical models. The structural identifiability (SI) assessment of nonlinear systems is one of the identification problems. Until now, this problem solves by parametric methods using various approximation methods. This approach is not always effective under uncertainty. We apply an approach to SI estimation based on the analysis of virtual framework. There is an objects class whose properties describe by nonsymmetric nonlinearities. The paper generalizes and develops the virtual framework (VF) method for systems with asymmetric non-linearities. Requirements for the system input are formed based on the excitation constancy property and S-synchronizability. Considering S-synchronizability gives VF that most fully reflect nonlinear properties of the system. A method for designing virtual structures based on the measurement information analysis describes. Structural identifiability fundamentals described for systems with symmetric nonlinearities. Splitting of the initial nonlinear system obtains for the VF application. Two methods consider for evaluating SI systems with nonsymmetric nonlinearities (NN) and propose their development on systems with nonsymmetric nonlinearities. Virtual framework almost homotheticity conditions obtain for SI estimation. A NN class with parametric features considers and conditions for estimating their almost homotheticity obtain. Conditions of almost homothety and h-identifiability obtain for systems with NN. The detectability and recoverability proofed for virtual frameworks guaranteed the SI estimation under uncertainty. The conditions under which the nonsymmetric nonlinearity is hypothetical symmetric nonlinearity obtained. The described approach to the SI assessment is general. If the SI of specific nonlinear systems analyzes, then features these systems consider. These features require modification of proposed algorithms and procedures. SI evaluation examples of closed nonlinear systems given under uncertainty and of the excitation constancy fulfillment.

On the Bahri–Lions conjecture for elliptic equations with non-symmetric nonlinearities

We prove the existence of infinitely many solutions for a class of elliptic Dirichlet problems with non-symmetric nonlinearities. In particular, in suitable domains of {mathbb {R}}^n with nge 3, this result gives a positive answer to a well known conjecture formulated by A. Bahri and P.L. Lions. The proof is based on a minimization method which does not require the use of techniques of deformation from the symmetry. This method allows us to piece together solutions of Dirichlet problems in suitable subdomains, so we obtain infinitely many nodal solutions with a prescribed nodal structure.

Infinitely many solutions for elliptic equations with non-symmetric nonlinearities

We deal with the existence of infinitely many solutions for a class of elliptic problems with non-symmetric nonlinearities. Our result, which is motivated by a well known conjecture formulated by A. Bahri and P.L. Lions, suggests a new approach to tackle these problems. The proof is based on a method which does not require to use techniques of deformation from the symmetry and may be applied to more general non-symmetric problems.

Adaptive Tracking Control of a Class of Constrained Euler–Lagrange Systems by Factorization of Dynamic Mass Matrix

Due to the uncertain parameters and/or the coupled matrices in a majority of Euler–Lagrange (EL) systems among multiple inputs and outputs, the controller designs for the constrained robots with unknown nonlinearities and disturbances are still challenging and difficult. In this paper, a new adaptive motion tracking control method for a class of constrained EL systems is presented. The main feature of the presented control is that high-dimensional vector-based integral Lyapunov function combined with a disturbance observer is presented for a class of EL systems with the nonsymmetric nonlinearity of input of the actuators. As long as the error trajectories deviate from or approach the sliding surface, it allows the disturbance estimation to adjust its value. The errors of tracking will converge to a small zone. Thus, stability of a closed-loop system can be ensured. When the designed parameters of the controller are chosen appropriately, the size of the tracking errors in stable state can be ensured. The applicability of this control method has been verified by the experiments with a planar robotic manipulator.

Nonlinear Dynamics and Control of a Pneumatic Vibration Isolator

Abstract The nonlinear dynamics of a single-degree-of-freedom pneumatic vibration isolator are studied. Based on a physical model, a nonsymmetric stiffness nonlinearity is derived to describe the stiffness property of the isolator. For a full nonlinear pneumatic isolator model, the response to two different types of disturbances is studied: forces applied to the isolated payload and base vibrations. The dynamic behavior of the isolator in case of a disturbance applied to the payload is studied using the generalized force-mobility function and features coexisting steady-state responses and a superharmonic resonance. Base vibrations transmitted via the isolator are studied on the basis of the generalized transmissibility function again showing rich nonlinear dynamic behavior. The presence of a nonsymmetric nonlinearity also induces high-energy low-frequency response to multiple high-frequency excitation. For both types of excitation, the nonlinear behavior is seriously compromising the performance of the isolator. To avoid any expression of nonlinearity whatsoever and, at the same time, to enhance the performance of the passive isolator, an overall nonlinear control design is proposed. It consists of a linear PID-based controller together with a nonlinear computed torque controller (CTC). For either linear or nonlinear control, the isolator performance is quantified in terms of generalized force mobility and transmissibility. The latter with a special focus on multiple high-frequency excitation.

Estimation of Non-Gaussian Response Distribution of a Nonlinear System Subjected to Random Excitation. Application to Nonwhite Excitation with Nonrational Spectrum.

The non-Gaussian response distribution of a single degree-of-freedom system with nonsymmetric nonlinearities subjected to nonwhite random excitation with nonrational spectra is investigated. The nonstationary and stationary probability densities of stochastic responses are calculated by employing the method which consists of the non-Gaussian equivalent linearization technique and the use of the moment equations approach. The non-Gaussian probability density function for the response distribution is expressed in terms of the potential of the system and employed for determining the equivalent linear coefficients. The results are compared with the corresponding digital simulation results.

Non-Gaussian Equivalent Linearization for Estimation of Stochastic Response Distribution of Nonlinear Systems.

A method is developed for estimating the response distribution of a nonlinear system subjected to nonwhite random excitation. The method consists of the non-Gaussian equivalent linearization technique and the use of the moment equations approach. The non-Gaussian probability density for the response distribution is expressed in terms of the potential of the system and employed for determining the equivalent linear coefficients. Numerical examples are given for a single degree-of freedom system with nonsymmetric nonlinearities. The rms and mean responses and the probability density of stochastic responses are obtained and compared with the corresponding digital simulation results.

Recursive expressions for time dependent means and mean square responses of a multi-degree-of-freedom non-linear system

In this paper we present recursive expressions for time dependent means and mean square responses of a multi-degree-of-freedom (MDOF) system with non-symmetric non-linearities in its stiffness terms. Based on the stochastic central difference-cum-statistical linearization (SCD-SL) technique, four recursive schemes for this system are derived. These schemes are implemented in a digital computer program written in Fortran 77. The results of both the stationary and time dependent means and mean square responses obtained by using the computer program are compared with those employing the Monte Carlo simulation (MCS) technique. It is concluded that (a) the recursive expression schemes are easy to be implemented in a digital computer program, (b) the computed stationary and time dependent means and mean square responses are accurate, and (c) the schemes are far more economical to employ than the MCS technique.

Transient response of a multi-span beam on non-symmetric non-linear supports

A multi-span beam on magnetic levitation supportssis of importance in various machines. The electromagnetic support has a strong non-symmetric non-linearity between the levitation force and the displacement. This paper is concerned with a method for analyzing the dynamic response of a multi-span beam on non-symmetric non-linear supports under an unsteady load decaying exponentially with time. In the analysis it is assumed that the non-linear spring system is a group of segments of linear springs, and the expression for the non-linear transient response is derived by use of a combination of Laplace transformation, Fourier series expansion and the residue theorem. Numerical calculations have been carried out for some sample problems, and the effects of non-linearity due to the non-linear supports have been illustrated.

A method of calculation of steady state oscillations in autonomous non-linear systems

An extension of the harmonic balance method is presented for calculation of the steady state (periodic) solutions in autonomous non-linear systems. The non-linear differential equation is expanded into a system of linear differential equations which are solved consecutively. An approximate solution which includes a few harmonics (the main oscillation) is found from the first equation of this system by the ordinary harmonic balance method. Other harmonics and the corrections of the main oscillation are found as the forced solutions of the second, third and the following equations of the system. To improve the efficiency of correction calculations a systematic method for derivation of the forcing term in these equations is given. As an example, a non-linear conservative system with a non-symmetric non-linearity of the polynomial type is considered.

Subharmonic oscillations in nonlinear rolling

In this paper the rolling motion of a ship is examined with particular regard to the possibility of obtaining oscillations which are subharmonic to the excitation frequency. Three different mechanisms are found to be responsible for this phenomenon, the importance of which has already been recognized in the context of ship stability. The first is related to a strong symmetric or non-symmetric nonlinearity in the righting arm. The second is linked to the harmonic composition of sea waves and the third to the well known parametric excitation caused by coupling between different ship motions in a following sea. The onset of subharmonics is related to a threshold value for the excitation strongly depending on damping. The more appropriate analytical methods for a theoretical study of each mechanism are suggested.

Non-stationary responses of a non-symmetric non-linear system subjected to a wide class of random excitation

The non-stationary variance and mean responses of a single-degree-of-freedom mechanical system with non-symmetric non-linearities subjected to a wide class of random excitation are investigated. For non-white excitation a modified equivalent linearization technique is proposed, in which the equivalent linear system is subjected to the equivalent forcing function with shifted mean. The results obtained by the present method are compared with the corresponding digital simulation results. The general form is provided for a multi-degree-of-freedom non-linear system with non-symmetric non-linearity.

Harmonic Analysis of Dynamic Systems With Nonsymmetric Nonlinearities

The generalized method of equivalent linearization is modified to be applicable for multi-degree-of-freedom dynamic systems with nonsymmetric nonlinearities subjected to harmonic monofrequency excitation. Readily applicable formulas are given for the construction of the equivalent linear systems related to a class of systems commonly encountered in engineering applications. As an example of its application, the proposed method is used to generate an approximate steady-state solution for a simple vehicle system. The accuracy of the approximate solutions is determined.