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This study investigates the nonlinear dynamics of a cracked plate undergoing large-amplitude vibration, aiming to show the influence of structural damage on the nonlinear characteristics of the plate. First, the governing equations of the cracked large-amplitude vibrating plate are derived by the von Karman theory, with the central part-through crack being simulated by a modified line-spring model for characterizing the crack-induced reduction in stress values. Second, the acquired partial differential equations are discretized by the Galerkin method combined with a simply supported boundary condition. Third, the harmonic balance method and the fourth-order Runge–Kutta algorithm are used to obtain the analytical and numerical approximations of the structural responses, respectively. For cases of small harmonic loads, the plate shows a hardening nonlinear frequency response. Influences of various parameters on this hardening nonlinearity are presented, including crack stage, aspect ratio, plate thickness, excitation location, and excitation amplitude. In cases of large harmonic loads, the bifurcations and chaotic dynamics of the cracked plate are explored by considering the effects of excitation amplitude and excitation frequency. Both analytical and numerical results indicate that a thick plate with a large aspect ratio is more sensitive to damage, especially when the crack parallels to the long side of the plate. Besides, the excitation that occurs far from the plate center with a large amplitude is beneficial for the damage characterization. Particularly, the damage-induced periodic-chaotic transition provides a novel insight into characterizing structural damage from perspectives of chaotic dynamics.

The work is devoted to study 2D pressure driven rarefied gas flow in a microchannel having an elastic obstacle. The elastic obstacle is clamped at the bottom channel wall and its length is half of the channel height. The gas flow is simulated by Direct Simulation Monte Carlo (DSMC) method applying the advanced Simplified Bernoulli Trial (SBT) collision scheme. The elastic obstacle is modelled as geometrically nonlinear Euler Bernoulli beam. A reduced 3 modes reduction model of the beam is created. The influence of the gas flow on the beam vibration is studied, considering the linear and nonlinear beam theories.

Fatigue damage in engineering structures is universal. The occurrence of fatigue cracks brings unpredictable hidden dangers to a structure in terms of safety and service performance. Traditional damage identification methods, such as power spectrum analysis, are mostly based on linear elasticity theory that cannot reflect the typical nonlinear characteristics of fatigue cracks and cannot meet the higher requirements of the signal analysis method put forward by current mass detection data. To solve this problem, a numerical model of a cantilever beam with a breathing crack is established in this study. A method for diagnosing fatigue damage is studied by combining bispectral analysis and a statistical normal cloud model, which characterize the nonlinear characteristics of the structure. This method can effectively describe the nonlinear characteristics of the structure and reasonably evaluate the degree of fatigue damage in the structure. The bispectrum-normal cloud model method proposed in this study overcomes the limitations of existing linear damage detection methods in nonlinear damage detection, and can improve the efficiency of signal analysis from a statistical point of view. It has good prospects for structural nonlinear damage assessment.

A nonlinear thermoelastic model of a circular plate is presented in the paper. The model, based on the Mindlin plate theory, is extended by taking into account nonlinear geometrical terms. Partial differential equations of plate’s dynamics are derived for a fully coupled thermal and mechanical fields. Then the model is reduced to a set of ordinary differential equations taking into account the first three natural modes and assuming a constant thermal field. The influence of elevated temperature on the resonance curves and the mode involvement due to nonlinear and thermal couplings is presented. The analysis shows that the increased temperature may lead to various bifurcation scenarios. The buckling phenomenon and post-buckling nonlinear regular and chaotic oscillations are studied.

Fatigue damage is a type of damage usually occurring to repeatedly loaded elements of structures in various engineering fields. Accumulation of fatigue damage may cause failure of structural elements. Identification of incipient fatigue damage is essential to ensure safety of structures. Fatigue crack under repeated loads commonly behaves in a nonlinear dynamic manner, typically manifested by both occurrence of higher harmonic components and interaction of harmonic components. Interrogation of nonlinear dynamic manner provides a promising way to characterize structural fatigue damage. This study aims at developing a new method to interrogate nonlinear dynamic manner for fatigue damage identification. This method is based on bispectral analysis of structural vibrational responses. This method portrays fatigue damage by inspecting the presence of higher harmonic components and quantifying the interaction of these harmonic components. The method can precisely locate and quantify a small-sized fatigue damage in a cantilever beam, presenting great accuracy in fatigue damage identification.

An effect of thermal loading on vibrations of beams is investigated in the paper. A beam is considered as an extended Timoshenko beam model with nonlinear terms resulted from large deflections. Dynamics of the structure is analysed under thermal and mechanical loadings considering transient dynamics due to a heat pulse imposed to the beam. The numerical method for solving coupled thermo-mechanical problem is presented. On this basis the importance of the heat pulse intensity around the first resonance condition is demonstrated. The effect of the heat on the the complex transient dynamics of the beam and its qualitatively different response is shown as well.

Nonlinear vibrations of a circular plate subjected to mechanical and thermal loadings are presented in the paper. A model of the plate is based on the extended Mindlin theory, taking into account nonlinear geometrical terms and acting heat uniformly distributed along the plate span. The dynamics of a coupled thermo-mechanical problem is reduced from a set of partial differential equations to ordinary differential equations. Considering oscillations around the first natural frequency just one mode reduction is proposed. The analysis shows that elevated temperature shifts the resonance curve and new post-buckling oscillations arise. Depending on initial conditions for the post-buckling state various scenarios of bifurcations take place and transient irregular oscillations may occur. The proposed one degree of freedom model shows a good agreement with response of the model based on three or five-modes reduction.

Dynamics of a Timoshenko beam under an influence of mechanical and thermal loadings is analysed in this paper. Nonlinear geometrical terms and a nonuniform heat distribution are taken into account in the considered model. The mathematical model is represented by a set of partial differential equations (PDEs) which takes into account thermal and mechanical loadings. The problem is simplified to two PDEs and then reduced to ordinary differential equations (ODEs) by means of the Galerkin method taking into account three modes of a linear Timoshenko beam. Correctness of the analytical model is verified by a finite element method. Then, the nonlinear model is studied numerically by a continuation method or by a direct numerical integration of ODEs. An effect of the temperature distribution on the resonance near the first natural frequency and on stability of the solutions is presented. The increase of mechanical loading results in hardening of the resonance curve. Thermal loading may stabilise the beam dynamics when the temperature is decreased. The elevated temperature may transit dynamics from regular to chaotic oscillations.

Large amplitude vibrations of a Timoshenko beam under an influence of temperature are analysed in this paper. In the considered model the temperature increases instantly and the heat is uniformly distributed along the beams length and cross-section. The mathematical model, represented by partial differential equations takes into account thermal and mechanical loadings. Next, the problem is reduced by means of the Galerkin method, considering the first three natural vibration modes of a simply supported beam in the both ends. The influence of the temperature on amplitudes and localisation of the resonance zones and stability of the solutions is studied numerically and analytically by the multiple time scale method. The bifurcation points, existence of unstable lobes and transition from regular to chaotic oscillations are shown.

This engineering research focus issue was inspired by the workshop organised by us on March 19th 2012 at the Lublin University of Technology (in the framework of CEMCAST) titled “Dynamics of functionally graded materials and systems with hysteresis” The collection of the present research papers followed our fruitful discussions focussing on “Dynamics of composite nonlinear systems and materials for engineering applications and energy harvesting” which is now the title of the present volume. In the next two sections we briefly discuss the topics included in this focus issue.

Recent NATO funded research on methods for detection and interpretation methodologies for damage detection in aircraft panel structures has motivated work on low-order nonlinear analytical modelling of vibrations in cracked isotropic plates, typically in the form of aluminium aircraft panels. The work applies fundamental aspects of fracture mechanics to define an elliptical crack, and the local stress field and loading conditions, arbitrarily located at some point in the plate, and then derives an analytical expression for this that can be incorporated into the PDE for an edge loaded plate with various possible boundary conditions. The plate PDE is converted into a nonlinear Duffing-type ODE in the time domain by means of a Galerkin procedure and then an arbitrarily small perturbation parameter is introduced into the equation in order to apply an appropriate solution method, in this case the method of multiple scales. This is used to solve the equation for the vibration in the cracked plate for the chosen boundary conditions, which, in turn, leads to an approximate analytical solution. The solution is discussed in terms of the perturbation approximations that have been applied and highlights the phenomenology inherent within the problem via the specific structures of the analytical solution.

The thermoelastic, geometrically nonlinear vibrations of isotropic, straight and curved beams are studied using p-version, hierarchical finite elements. First-order shear deformation theory is followed, and both the longitudinal displacements and inertia are taken into account. The nonlinear equations of motion are solved in the time domain by Newmark's method. The influence of parameters like the temperature variation, the thickness and the ratio of curvature on the beams nonlinear dynamics is studied. Periodic and non-periodic motions are found.

Large amplitude vibrations of a Timoshenko beam under an influence of thermal and mechanical loadings are studied in the paper. The structural parameters of the beam are considered enabling internal resonance conditions. Moreover, it is assumed that the beam gets instantly temperature which is distributed along its length and thickness. The mathematical model represented by a set of partial differential equations takes into account coupled mechanical and thermal fields. The problem is transformed to a set of ODEs by the Galerkin method and three modes of a simply supported beam at both ends are studied. The effect of temperature on internal and external resonances is analysed on the basis of the proposed reduced model.Copyright © 2014 by ASME

The goal of this paper is to study large amplitude vibrations of a Timoshenko beam under an influence of the elevated temperature. It is assumed that the beam gets the elevated temperature instantly and the temperature is uniformly distributed along the beam’s length and cross-section. The mathematical model represented by a set of partial differential equations is derived taking into account boundary conditions for a simply supported beam in the both ends. Next, the problem is reduced by the Galerkin method by means of free vibration modes. The influence of the temperature on a resonance localisation and nonlinear oscillations is studied numerically and analytically by the multiple time scale method.

In this work, the geometrically nonlinear vibrations of fully clamped composite beams subjected to thermal changes are used to study the sensitivity of selected vibration response parameters to the presence of delamination (damage) and elevated temperature. The damage detection criterion formulated earlier for non-heated plates, based on analyzing the points in the Poincare sections of the damaged and healthy plate, is modified and tested for the case of beams additionally subjected to elevated temperatures. The importance of the actual temperature in the process of damage detection is shown.

In this work, geometrically nonlinear vibrations of fully clamped rectangular plates subjected to thermal changes are used to study the sensitivity of some vibration response parameters to the presence of damage and elevated temperature. The geometrically nonlinear version of the Mindlin plate theory is used to model the plate behaviour. Damage is represented as a thickness reduction in a small area of the plate. The plates are subjected to harmonic loading leading to large amplitude vibrations and temperature changes. The plate vibration response is obtained by a pseudo-load mode superposition method. The main results are focussed on establishing the influence of damage on the vibration response of the heated and the unheated plates and the change in the time-history diagrams and the Poincare maps caused by damage and elevated temperature. The damage criterion formulated earlier for non-heated plates, based on analyzing the points in the Poincare sections of the damaged and healthy plate, is modified and tested for the case of plates additionally subjected to elevated temperatures. The importance of taking into account the actual temperature in the process of damage detection is shown.

The dynamic contact interaction between an elastic-plastic beam with a variable cross-section and a two-parametric elastic foundation with a finite thickness is studied. The beam is subjected to an impulsive loading and its plastic properties are described by a linear isotropic strain-hardening model. The finite difference approximation is applied to the spatial domain and the fourth-order Runge-Kutta method is used to solve the nonlinear problem in the time domain. The development of the contact spot in time and the influence of the contact forces and foundation properties on the elastic and elastic-plastic beam response are presented.

This study aimed at the development of vibration-based health monitoring methodology for thin circular plates. The possibility of using the first several natural frequencies of a circular plate for damage detection purposes was investigated first. The study then suggested a damage detection method, which considered a vibrating plate as a dynamic system and used its time-domain response represented in a new phase (state) space to extract damage sensitive characteristics. The paper introduced the idea of using large amplitude vibrations and nonlinear time series analysis for damage detection purposes. The suggested damage detection approach explored the possibility to use certain characteristics of the distribution of phase space points on the attractor of the system. It studied the histograms of this distribution and attempts to extract damage sensitive features. Three damage features were suggested and they are shown to detect damage at a rather low level using a finite element model of the plate. The method suggested was rather generic and permits development and application to more complex structures and real data.

This study proposes an analytical model for vibrations in a cracked rectangular plate as one of the results from a program of research on vibration based damage detection in aircraft panel structures. This particular work considers an isotropic plate, typically made of aluminum, and containing a crack in the form of a continuous line with its center located at the center of the plate and parallel to one edge of the plate. The plate is subjected to a point load on its surface for three different possible boundary conditions, and one examined in detail. Galerkin’s method is applied to reformulate the governing equation of the cracked plate into time dependent modal coordinates. Nonlinearity is introduced by appropriate formulations introduced by applying Berger’s method. An approximate solution technique—the method of multiple scales—is applied to solve the nonlinear equation of the cracked plate. The results are presented in terms of natural frequency versus crack length and plate thickness, and the nonlinear amplitude response of the plate is calculated for one set of boundary conditions and three different load locations, over a practical range of external excitation frequencies.

This paper deals with the problem of vibration health monitoring (VHM) in structures with nonlinear dynamic behaviour. It aims to introduce two viable VHM methods that use large amplitude vibrations and are based on nonlinear time series analysis. The methods suggested explore some changes in the state space geometry/distribution of the structural dynamic response with damage and their use for damage detection purposes. One of the methods uses the statistical distribution of state space points on the attractor of a vibrating structure, while the other one is based on the Poincare map of the state space projected dynamic response. In this paper both methods are developed and demonstrated for a thin vibrating plate. The investigation is based on finite element modelling of the plate vibration response. The results obtained demonstrate the influence of damage on the local dynamic attractor of the plate state space and the applicability of the proposed strategies for damage assessment. The approach taken in this study and the suggested VHM methods are rather generic and permit development and applications for other more complex nonlinear structures.

Dynamics of a composite beam subjected to thermal and mechanical loadings is presented in the paper. The extended Timoshenko beam model takes into account shear and inertia of the cross-section and nonlinear longitudinal displacement caused by mechanical and thermal loadings. It has been shown that thermal and mechanical fields are fully coupled and the heat pulse may change the transient dynamics of the system. For the case of elevated temperature and a steady state problem the model is reduced to nonlinear ordinary differential equations of motion. It has been shown that the elevated ambient temperature may drastically change its response.