A new method for the non-linear deflection analysis of an infinite beam resting on a non-linear elastic foundation

The aim of this paper is to develop a new method of analyzing the non-linear deflection behavior of an infinite beam on a non-linear elastic foundation. Non-linear beam problems have traditionally been dealt with by semi-analytical approaches that involve small perturbations or by numerical methods, such as the non-linear finite element method. In this paper, in contrast, a transformed non-linear integral equation that governs non-linear beam deflection behavior is formulated to develop a new method for non-linear solutions. The proposed method requires an iteration to solve non-linear problems, but is fairly simple and straightforward to apply. It also converges quickly, whereas traditional non-linear solution procedures are generally quite complex in application. Mathematical analysis of the proposed method is performed. In addition, illustrative examples are presented to demonstrate the validity of the method developed in the present study.     

A computationally efficient non-linear beam model

Aerospace structures with large aspect ratio, such as airplane wings, rotorcraft blades, wind turbine blades, and jet engine fan and compressor blades, are particularly susceptible to aeroelastic phenomena. Finite element analysis provides an effective and generalized method to model these structures; however, it is computationally expensive. Fortunately, the large aspect ratio of these structures is exploitable as these potential aeroelastically unstable structures can be modeled as cantilevered beams, drastically reducing computational time.In this paper, the non-linear equations of motion are derived for an inextensional, non-uniform cantilevered beam with a straight elastic axis. Along the elastic axis, the cross-sectional center of mass can be offset in both dimensions, and the principal bending and centroidal axes can each be rotated uniquely. The Galerkin method is used, permitting arbitrary and abrupt variations along the length that require no knowledge of the spatial derivatives of the beam properties. Additionally, these equations consistently retain all third-order non-linearities that account for flexural-flexural-torsional coupling and extend the validity of the equations for large deformations.Furthermore, linearly independent shape functions are substituted into these equations, providing an efficient method to determine the natural frequencies and mode shapes of the beam and to solve for time-varying deformation.This method is validated using finite element analysis and is extended to swept wings. Finally, the importance of retaining cubic terms, in addition to quadratic terms, for non-linear analysis is demonstrated for several examples.     

An analytic solution of transversal oscillation of quintic non-linear beam with homotopy analysis method

non-linear vibration analysis of beam used in steel structures is of particular importance in mechanical and industrial applications. To achieve a proper design of the beam structures, it is essential to realize how the beam vibrates in its transverse mode which in turn yields the natural frequency of the system. Equation of transversal vibration of hinged–hinged flexible beam subjected to constant excitation at its free end is identified as a non-linear differential equation. The quintic non-linear equation of motion is derived based on Hamilton’s principle and solved by means of an analytical technique, namely the Homotopy analysis method. To verify the soundness of the results, a comparison between analytical and numerical solutions is developed. Finally, to express the impact of the quintic nonlinearity, the non-linear responses obtained by HAM are compared with the results from usual beam theory.     

A complete boundary integral formulation for steady compressible inviscid flows governed by non-linear equations

A complete boundary integral formulation for steady compressible inviscid flows governed by non-linear equations is established by using the specific mass flux as a dependent variable. Thus, the dimensionality of the problem to be solved is reduced by one and the computational mesh to be generated is needed only on the boundary of the domain. It is shown that the boundary integral formulation developed in this paper is equivalent of the results of distributions of the fundamental solutions of the Laplacian operator equation with a different order along the boundaries of the domain. Hence, we have succeeded in establishing the fundamental-solution method for compressible inviscid flows governed by non-linear equations.     

Direct method for analysis of flexible cantilever beam subjected to two follower forces

The static analysis of the flexible non-uniform cantilever beams under a tip-concentrated and intermediate follower forces is considered. The angles of inclination of the concentrated forces with respect to the deformed axis of the beam remain unchanged during deformation. The governing non-linear boundary-value problem is reduced to an initial-value problem by change of variables. The resulting problem can be solved without iterations. It is shown that there are no critical loads in the Euler sense (divergence) for any flexural-stiffness distribution and angles of inclination of the follower forces. In particular, if the follower forces are tangential, the rectilinear shape of the non-uniform cantilever beam is the only possible equilibrium configuration. In this paper some equilibrium configurations of the uniform cantilever under normal or tangential follower forces are presented using direct method.     

A new method for detecting non-linear damping and restoring forces in non-linear oscillation systems from transient data

The purpose of this study is to recover the functional form of both non-linear damping and non-linear restoring forces in the non-linear oscillatory motions of an autonomous system. Using two sets of measured motion response data of the system, an inverse problem is formulated for recovering (or identification): the differential equation of motion is transformed into an equivalent integral equation of motion. The identification, which is non-linear, is shown to be one-to-one. However, the inverse problem formulated herein is concerned with the Volterra-type of non-linear integral equation of the first kind. This leads to numerical instability: solutions of the inverse problem lack stability properties. In order to overcome the difficulty, a regularization method is applied to the identification process. In addition, an L-curve criterion, combined with regularization, is introduced to find an optimal choice for the regularization parameter (i.e., the number of iterations), in the presence of noisy data. The workability of the identification is investigated for simultaneously recovering the functional form of the non-linear damping and the non-linear restoring forces through a numerical experiment.     

A novel statistical linearization solution for randomly excited coupled bending-torsional beams resting on non-linear supports

A novel statistical linearization technique is developed for computing stationary response statistics of randomly excited coupled bending-torsional beams resting on non-linear elastic supports. The key point of the proposed technique consists in representing the non-linear coupled response in terms of constrained linear modes. The resulting set of non-linear equations governing the modal amplitudes is then replaced by an equivalent linear one via a classical statistical error minimization procedure, which provides algebraic non-linear equations for the second-order statistics of the beam response, readily solved by a simple iterative scheme. Data from Monte Carlo simulations, generated by a pertinent boundary integral method in conjunction with a Newmark numerical integration scheme, are used as benchmark solutions to check accuracy and reliability of the proposed statistical linearization technique.

     

Application of vector-valued rational approximations to a class of non-linear oscillations

By combining a perturbation technique with a rational approximation of vector-valued function, we propose a new approach to non-linear oscillations of conservative single-degree-of-freedom systems with odd non-linearity. The equation of motion does not require to contain a small parameter. First, the Lindstedt-Poincare perturbation method is used to obtain an asymptotic analytical solution. Then the range of validity of the analytical representation is extended by using the vector-valued rational approximation of functions. For constructing the rational approximations, all that is needed is the coefficients of the perturbation expansion being considered. General approximate formulas for period and the corresponding periodic solution of a non-linear system are established. Two examples are used to illustrate the effectiveness of the proposed method.     

On the use of non-linear vibrations and the anti-resonances of Higher-Order Frequency Response Functions for crack detection in pipeline beam

The identification of new scientific challenges, as well as the increasing high-performance computing support, indicates that the benefits of applying novel nonlinear techniques for crack detection will continue to grow. So, significant effort has been invested in recent years to develop effective techniques to detect crack in mechanical structures. The objective of this paper is to discuss and propose a robust diagnostic of damage based on non-linear vibrational measurements with particular regard to the Higher-Order Frequency Response Functions. An important observation is that the appearances of the non-linear harmonic components and the emerging anti-resonances in Higher-Order Frequency Response Functions can provide useful information on the presence of cracks and may be used on an on-line crack monitoring system for small levels of damage. Efficiency of the proposed methodology is illustrated through numerical examples for a pipeline beam including a breathing crack.     

On a non-linear moving boundary problem for a diffusion–convection equation

We study a one-dimensional free boundary problem for a non-linear diffusion–convection equation whose diffusivity is heterogeneous in space as well as being non-linear. Under the Bäcklund transformation the problem is reduced to an associated free boundary problem. We prove the existence and uniqueness, local in time, of the solution by using the Friedman Rubinstein integral representation method and the Banach contraction theorem.     

An investigation into the significance of the non-linear terms in the equations of motion for a cantilevered beam

Non-linear beam models can provide accurate and efficient results and are therefore used frequently in the structural dynamic analysis of large-aspect-ratio cantilevered structures. Using a beam model is especially beneficial in aeroelastic applications, which are computationally expensive.This paper investigates the significance of each of the non-linear terms in a third-order non-linear beam model and shows the importance of retaining terms up to the third order. The approach presented in this paper provides an effective method to simplify the non-linear equations of motion by including the most significant non-linear terms. The resulting equations preserve the fidelity of the original equations and are easier to manipulate.     

Generic non-linear modelling of a bi-material composite beam with partial shear interaction

Composite members composed of two materials joined by shear connection find widespread use in engineering infrastructure, in both traditional practice and innovative applications. Studies in the literature dating back nearly 60 years have elucidated the mechanics of the behaviour of these composite structural members in which the solution for the slip at the interface between the materials was determined by solving a linear differential equation. However, these solutions are based on a linear formulation of the strain-displacement relationship, and in some applications this relationship must be represented in non-linear form, so that the second order effects in the member can be quantified correctly. This paper presents such a study for a composite member with two materials, being typical of a steel-concrete composite beam in structural engineering. It quantifies the restraint of the member ends by longitudinal and rotational elastic springs, so that the axial tension developed is a function of the transverse loading, material properties, cross-sectional properties and the restraint stiffness. The problem is treated using minimisation of the total potential stored in the two members, the elastic shear connection at their interface, the restraints at the ends and the work done by the transverse forces, for which the differential equations for the deformations can be determined from routine variational calculus. The non-linear equation of equilibrium relating the external loading to the internal actions is stated in closed form by invoking the static and kinematic boundary conditions for the member. The solution is compared with closed form treatments derived elsewhere, and a representative member is analysed so that the influences of the non-linearity, end restraint stiffness and degree of partial shear interaction on its behaviour can be examined.     

一个改进的平面梁单元

根据有限单元法基本原理 ,提出了一个变截面平面梁单元 ,推导了其单元钢度矩阵。这一改进的梁单元用于分析梁高呈线性变化及二次抛物线变化的矩形截面梁 ,将得到准确解。文中给出了一个变截面悬臂梁算例 ,计算表明 ,这一改进的梁单元使变截面梁的分析大大简化     

Non-linear response of a beam to periodic loading

The steady state response of a non-linear beam under periodic excitation is investigated. The non-linearity is attributed to the membrane tension effect which is induced in the beam when the deflection is not small in comparison to its thickness. The effects of multimode participation are investigated for simply supported and clamped boundary conditions. The finite element technique is used to formulate the non-linear differential equations of the straight beam and the method of averaging is used to obtain an approximate solution to the non-linear equations under harmonic loading. An analog computer was used to simulate the non-linear beam equation which was subjected to harmonic excitation. The agreement between theoretical and experimental values is reasonably good.     

The non-linear dynamic response of the Euler-Bernoulli beam with an arbitrary number of switching cracks

In this study the non-linear dynamic response of the Euler-Bernoulli beam in presence of multiple concentrated switching cracks (i.e. cracks that are either fully open or fully closed) is addressed. The overall behaviour of such a beam is non-linear due to the opening and closing of the cracks during the dynamic response; however, it can be regarded as a sequence of linear phases each of them characterised by different number and positions of the cracks in open state. In the paper the non-linear response of the beam with switching cracks is evaluated by determining the exact modal properties of the beam in each linear phase and evaluating the corresponding time history linear response through modal superposition analysis. Appropriate initial conditions at the instant of transition between two successive linear phases have been considered and an energy control has been enforced with the aim of establishing the minimum number of linear modes that must be taken into account in order to obtain accurate results. Some numerical applications are presented in order to illustrate the efficiency of the proposed approach for the evaluation of the non-linear dynamic response of beams with multiple switching cracks. In particular, the behaviour under different boundary conditions both for harmonic loading and free vibrations has been investigated.     

A non-linear uniaxial integral constitutive equation incorporating rate effects,creep and relaxation

A previously proposed first order non-linear differential equation for uniaxial viscoplasticity, which is non-linear in stress and strain but linear in stress and strain rates, is transformed into an equivalent integral equation. The proposed equation employs total strain only and is symmetric with respect to the origin and applies for tension and compression. The limiting behavior for large strains and large times for monotonic, creep and relaxation loading is investigated and appropriate limits are obtained. When the equation is specialized to an overstress model it is qualitatively shown to reproduce key features of viscoplastic behavior. These include: initial linear elastic or linear viscoelastic response: immediate elastic slope for a large instantaneous change in strain rate normal strain rate sensitivity and non-linear spacing of the stress-strain curves obtained at various strain rates; and primary and secondary creep and relaxation such that the creep (relaxation) curves do not cross. Isochronous creep curves are also considered. Other specializations yield wavy stress-strain curves and inverse strain rate sensitivity. For cyclic loading the model must be modified to account for history dependence in the sense of plasticity.     

Stochastic non-linear response of a flexible rotor with local non-linearities

The effects of uncertainties on the non-linear dynamics response remain misunderstood and most of the classical stochastic methods used in the linear case fail to deal with a non-linear problem. So we propose to take into account of uncertainties into non-linear models, by coupling the Harmonic Balance Method (HBM) and the Polynomial Chaos Expansion (PCE). The proposed method called the Stochastic Harmonic Balance Method (Stochastic-HBM) is based on a new formulation of the non-linear dynamic problem in which not only the approximated non-linear responses but also the non-linear forces and the excitation pulsation are considered as stochastic parameters. Expansions on the PCE basis are performed by passing via an Alternate Frequency Time method with Probabilistic Collocation (AFTPC) for estimating the stochastic non-linear forces in the stochastic domain and the frequency domain. In the present paper, the Stochastic Harmonic Balance Method (Stochastic-HBM) that is applied to a flexible non-linear rotor system, with random parameters modeled as random fields, is presented. The Stochastic-HBM combined with an Alternate Frequency-Time method with Probabilistic Collocation (AFTPC) allows us to solve dynamical problems with non-regular non-linearities in presence of uncertainties. In this study, the procedure is developed for the estimation of stochastic non-linear responses of the rotor system with different regular and non-regular non-linearities. The finite element rotor system is composed of a shaft with two disks and two flexible bearing supports where the non-linearities are due to a radial clearance or a cubic stiffness. A numerical analysis is performed to analyze the effect of uncertainties on the non-linear behavior of this rotor system by using the Stochastic-HBM. Furthermore, the results are compared with those obtained by applying a classical Monte-Carlo simulation to demonstrate the efficiency of the proposed methodology.     

Transient elastic wave propagation in an infinite Timoshenko beam on viscoelastic foundation

This paper presents an effective numerical method for solving elastic wave propagation problems in an infinite Timoshenko beam on viscoelastic foundation in time domain. In order to use the finite element method to model the local complicated material properties of the infinite beam as well as foundation, two artificial boundaries are needed in the infinite system so as to truncate the infinite beam into a finite beam. This treatment requires an appropriate boundary condition derived and applied on the corresponding truncated boundaries. For this purpose, the time-dependent equilibrium equation of motion for beam is changed into a linear ordinary differential equation by using the operator splitting and the residual radiation methods. Simultaneously, an artificial parameter is employed in the derivation. As a result, the high-order accurate artificial boundary condition, which is local in time, is obtained by solving the ordinary differential equation. The numerical examples given in this paper demonstrate that the proposed method is of high accuracy in dealing with elastic wave propagation problems in an infinite foundation beam.     

Inflation of a bonded non-linear viscoelastic toroidal membrane

The inflation of a bonded viscoelastic toroidal membrane under finite deformations is considered. Three new variables, viz. the two principal stretch ratios and the angle between the normal vector of a deformed membrane and the axis of symmetry are introduced as dependent variables. The governing equations are reduced thereafter to a set of three first-order partial differential integral equations. The constitutive equation developed by Pipkin and Rogers for the non-linear response of a viscoelastic material is used. The creep phenomenon for an inflated viscoelastic toroidal membrane under a constant pressure is presented.     

A perturbation method for certain non-linear oscillators

We present a perturbation method for the analysis of single degree of freedom non-linear oscillation phenomena governed by an equation of motion containing a parameter ? which need not be small. The approach is to define a new parameter α = α(?) in such a way that asymptotic solutions in power series in α converge more quickly than do the standard perturbation expansions in power series in ?. Phenomena considered are free vibration of strongly non-linear conservative oscillators and steady state response of strongly non-linear oscillators subject to weak harmonic excitation.