In this stud y, we attempt to analyse free nonlinear vibrations of buckling in laminated composite beams. Two new methods are applied to obtain the analytical solution of the nonlinear governing equation of the problem. The effects of different parameters on the ratio of nonlinear to linear natural frequencies of the beams are studied. These methods give us an agreement with numerical results for the whole range of the oscillation amplitude.
The nonlinear interaction of wave and non-wave modes in a gas planar flow are considered. Attention is mainly paid to the case when one sound mode is dominant and excites the counter-propagating sound mode and the entropy mode. The modes are determined by links between perturbations of pressure, density, and fluid velocity. This definition follows from the linear conservation equations in the differential form and thermodynamic equations of state. The leading order system of coupling equations for interacting modes is derived. It consists of diffusion inhomogeneous equations. The main aim of this study is to identify the principle features of the interaction and to establish individual contributions of attenuation (mechanical and thermal attenuation) in the solution to the system.
The object of the present study is to investigate the influence of damping uncertainty and statistical correlation on the dynamic response of structures with random damping parameters in the neighbourhood of a resonant frequency. A Non-Linear Statistical model (NLSM) is successfully demonstrated to predict the probabilistic response of an industrial building structure with correlated random damping. A practical computational technique to generate first and second-order sensitivity derivatives is presented and the validity of the predicted statistical moments is checked by traditional Monte Carlo simulation. Simulation results show the effectiveness of the NLSM to estimate uncertainty propagation in structural dynamics. In addition, it is demonstrated that the uncertainty in damping indeed influences the system response with the effects being more pronounced for lightly damped structures, higher variability and higher statistical correlation of damping parameters.
The paper presents a method of how the nonlinear boundary condition  may be applied in nonlinear problems of electromagnetic field theory. It is introduced for problems with nonlinear conductivity. An analytical procedure has been constructed, which seeks to reduce calculations related with the nonlinear region. In order to verify the proposed solutions, two problems have been formulated: one of linear and the other of cylindrical symmetry. These have been additionally solved by the authors’ modification of the perturbation method that has been described in previous papers [7, 8, 10]. The electromagnetic field distribution obtained thereby has served as a referential result since it can obtain very accurate solutions . Relative errors of electric and magnetic field strength are introduced to verify the results.
The task of electroacoustic devices is a transmission of audio signals. The transmitted signal should be distorted as little as possible. Nonlinear distortions are the distortions depending on signal level. The types of nonlinear distortions as well as their measures are presented in the paper. The weakest device in an electroacoustic chain is a loud-speaker. It causes the greatest degradation of the signal. It is usually the most nonlinear part of the electroacoustic system. The nonlinearities in loudspeakers are described in details. Other types of nonlinear distortions as transient intermodulation in power amplifiers and distortions caused by the A/C sampling are also presented.
Professor Piotr Pierański, an outstanding Polish physicists, excellent researcher and brilliant lecturer, passed away on the 23rd February 2018. The article quotes some recollections of his numerous friends and coworkers wordwide.
The positivity and absolute stability of a class of nonlinear continuous-time and discretetime systems are addressed. Necessary and sufficient conditions for the positivity of this class of nonlinear systems are established. Sufficient conditions for the absolute stability of this class of nonlinear systems are also given.
The presence of more than one solute diffused in fluid mixtures is very often requested for discussing the natural phenomena such as transportation of contaminants, underground water, acid rain and so on. In the paper, the effect of nonlinear thermal radiation on triple diffusive convective boundary layer flow of Casson nanofluid along a horizontal plate is theoretically investigated. Similarity transformations are utilized to reduce the governing partial differential equations into a set of nonlinear ordinary differential equations. The reduced equations are numerically solved using Runge-Kutta-Fehlberg fourth-fifth order method along with shooting technique. The impact of several existing physical parameters on velocity, temperature, solutal and nanofluid concentration profiles are analyzed through graphs and tables in detail. It is found that, modified Dufour parameter and Dufour solutal Lewis number enhances the temperature and solutal concentration profiles respectively.
This paper presents the design of digital controller for longitudinal aircraft model based on the Dynamic Contraction Method. The control task is formulated as a tracking problem of velocity and flight path angle, where decoupled output transients are accomplished in spite of incomplete information about varying parameters of the system and external disturbances. The design of digital controller based on the pseudo-continuous approach is presented, where the digital controller is the result of continuous-time controller discretization. A resulting output feedback controller has a simple form of a combination of low-order linear dynamical systems and a matrix whose entries depend nonlinearly on certain known process variables. Simulation results for an aircraft model confirm theoretical expectations.
The problem of mathematical modelling and indication of properties of a DIP has been investigated in this paper. The aim of this work is to aggregate the knowledge on a DIP modelling using the Euler-Lagrange formalism in the presence of external forces and friction. To indicate the main properties important for simulation, model parameters identification and control system synthesis, analytical and numerical tools have been used. The investigated properties include stability of equilibrium points, a chaos of dynamics and non-minimum phase behaviour around an upper position. The presented results refer to the model of a physical (constructed) DIP system.
This paper analyses the influence of the applied microwave power output on the intensification of drying in the context of process kinetics and product quality. The study involved testing samples of beech wood (Fagus sylvatica L.). Wood samples were dried in the microwave chamber at: 168 W, 210 W, 273 W, 336 W and 378 W power output level. For comparison, wood was dried convectively at 40 ◦C and 87% air relative humidity. The analysis of drying process kinetics involved nonlinear regression employing the Gompertz model. Dried samples were subjected to static bending tests in order to specify the influence of the applied microwave power on modulus of elasticity (MOE) and modulus of rapture (MOR). The obtained correlations of results were verified statistically. Analysis of drying kinetics, strength test results and Tukey’s test showed that the applied microwaves of a relatively low level significantly shortened the drying time, but did not cause a reduction in the final quality of dried wood, compared with conventional drying.
The study makes an attempt to model a complete vibrating guitar including its non-linear features, specifically the tension-compression of truss rod and tension of strings. The purpose of such a model is to examine the influence of design parameters on tone. Most experimental studies are flawed by uncertainties introduced by materials and assembly of an instrument. Since numerical modelling of instruments allows for deterministic control over design parameters, a detailed numerical model of folk guitar was analysed and an experimental study was performed in order to simulate the excitation and measurement of guitar vibration. The virtual guitar was set up like a real guitar in a series of geometrically non-linear analyses. Balancing of strings and truss rod tension resulted in a realistic initial state of deformation, which affected the subsequent spectral analyses carried out after dynamic simulations. Design parameters of the guitar were freely manipulated without introducing unwanted uncertainties typical for experimental studies. The study highlights the importance of acoustic medium in numerical models.
Excitation of the entropy mode in the field of intense sound, that is, acoustic heating, is theoretically considered in this work. The dynamic equation for an excess density which specifies the entropy mode, has been obtained by means of the method of projections. It takes the form of the diffusion equation with an acoustic driving force which is quadratically nonlinear in the leading order. The diffusion coefficient is proportional to the thermal conduction, and the acoustic force is proportional to the total attenuation. Theoretical description of instantaneous heating allows to take into account aperiodic and impulsive sounds. Acoustic heating in a half-space and in a planar resonator is discussed. The aim of this study is to evaluate acoustic heating and determine the contribution of thermal conduction and mechanical viscosity in different boundary problems. The conclusions are drawn for the Dirichlet and Neumann boundary conditions. The instantaneous dynamic equation for variations in temperature, which specifies the entropy mode, is solved analytically for some types of acoustic exciters. The results show variation in temperature as a function of time and distance from the boundary for different boundary conditions.
A navigation complex of an unmanned flight vehicle of small class is considered. Increasing the accuracy of navigation definitions is done with the help of a nonlinear Kalman filter in the implementation of the algorithm on board an aircraft in the face of severe limitations on the performance of the special calculator. The accuracy of the assessment depends on the available reliable information on the model of the process under study, which has a high degree of uncertainty. To carry out high-precision correction of the navigation complex, an adaptive non-linear Kalman filter with parametric identification was developed. The model of errors of the inertial navigation system is considered in the navigation complex, which is used in the algorithmic support. The procedure for identifying the parameters of a non-linear model represented by the SDC method in a scalar form is used. The developed adaptive non-linear Kalman filter is compact and easy to implement on board an aircraft.
This paper presents a robust model free controller (RMFC) for a class of uncertain continuous-time single-input single-output (SISO) minimum-phase nonaffine-in-control systems. Firstly, the existence of an unknown dynamic inversion controller that can achieve control objectives is demonstrated. Afterwards, a fast approximator is designed to estimate as best as possible this dynamic inversion controller. The proposed robust model free controller is an equivalent realization of the designed fast approximator. The perturbation theory and Tikhonov’s theorem are used to analyze the stability of the overall closed-loop system. The performance of the developped controller are verified experimentally in the position control of a pneumatic actuator system.
In recent years, with the rapid development of digital components, digital electronic computers, especially microprocessors, digital controllers have replaced analog controllers on many occasions. The application of digital controller makes the performance analysis of impulsive system more and more important. This paper considers global exponential stability (GES) of impulsive delayed nonlinear hybrid differential systems (IDNHDS).Through the application of the Lyapunov method and the Razumikhin technique, a series of uncomplicated and useful guiding principles have been obtained. The results of a numerical simulation are presented to demonstrate that the method is correct and effective.
The acoustic properties of the sitar string are studied with the aid of a physical model. The nonlinearity of the string movement caused by the bridge acting as an obstacle to the vibrating string is of special interest. Comparison of the model's audio output to recordings of the instrument shows interesting similarities. The effects dispersion and bridge have on the sound of the instrument are demonstrated in the model.
A heterogeneous Bertrand duopoly game with bounded rational and adaptive players manufacturing differentiated products is subject of investigation. The main goal is to demonstrate that participation of one bounded rational player in the game suffices to destabilize the duopoly. The game is modelled with a system of two difference equations. Evolution of prices over time is obtained by iteration of a two dimensional nonlinear map. Equilibria are found and local stability properties thereof are analyzed. Complex behavior of the system is examined by means of numerical simulations. Region of stability of the Nash equilibrium is demonstrated in the plane of the speeds of adjustment. Period doubling route to chaos is presented on the bifurcation diagrams and on the largest Lyapunov characteristic exponent graph. Lyapunov time is calculated. Chaotic attractors are depicted and their fractal dimensions are computed. Sensitive dependence on initial conditions is evidenced.
The paper focuses on different approaches to the safety assessment of concrete structures designed using nonlinear analysis. The method based on the concept of partial factors recommended by Eurocodes, and methods proposed by M. Holicky, and by the author of this paper are presented, discussed and illustrated on a numerical example. Global safety analysis by M. Holicky needs estimation of the resistance coefficient of variation from the mean and characteristic values of resistance, and requires two separate nonlinear analyses. The reliability index value and the sensitivity factor for resistance should be also identified. In the method proposed in this paper, the resistance coefficient of variation necessary to calculate the characteristic value of resistance may be adopted from test results and the resultant partial factor for materials properties, and may be calculated according to Eurocodes. Thus, only one nonlinear analysis from mean values of reinforcing steel and concrete is required.
The global (absolute) stability of nonlinear systems with negative feedbacks and positive descriptor linear parts is addressed. Transfer matrices of positive descriptor linear systems are analyzed. The characteristics u = f(e) of the nonlinear parts satisfy the condition k₁e ≤ f(e) ≤ k₂e for some positive k₁, k₂. It is shown that the nonlinear feedback systems are globally asymptotically stable if the Nyquist plots of the positive descriptor linear parts are located in the right-hand side of the circles (–¹/k₁, –¹/k₂).
This paper presents experimental observation of nonlinear vibrations in the response of a flexible cantilever beam to transverse harmonic base excitations around its flexural mode frequencies. In the experimental setup, instead of manual control of the signal excitation frequency and amplitude, a closed-loop vibration system is used to keep the excitation amplitude constant during the frequency sweep and to increase confidence in the experimental results. The experimental results show the presence of the third mode in the response when varying the excitation frequency around the fourth mode. The frequency-response curves, response spectrum and Poincaré plots were used for characterization of nonlinear dynamic behaviour of the beam.
The aim of this paper is to show that a real order generalization of the dissipative concepts is a useful tool to determine the stability (in the Lyapunov and in the input-output sense) and to design control strategies not only for fractional order non-linear systems, but also for systems composed of integer and fractional order subsystems (mixed-order systems). In particular, the fractional control of integer order system (e.g. PIλ control) can be formalized. The key point is that the gradations of dissipativeness, passivity and positive realness concepts are related among them. Passivating systems is used as a strategy to stabilize them, which is studied in the non-adaptive as well as in the adaptive case.