Chaotic vibrations of a beam with non-linear boundary conditions

Forced vibrations of an elastic beam with non-linear boundary conditions are shown to exhibit chaotic behavior of the strange attractor type for a sinusoidal input force. The beam is clamped at one end, and the other end is pinned for the tip displacement less than some fixed value and is free for displacements greater than this value. The stiffness of the beam has the properties of a bi-linear spring. The results may be typical of a class of mechanical oscillators with play or amplitude constraining stops. Subharmonic oscillations are found to be characteristic of these types of motions. For certain values of forcing frequency and amplitude the periodic motion becomes unstable and nonperiodic bounded vibrations result. These chaotic motions have a narrow band spectrum of frequency components near the subharmonic frequencies. Digital simulation of a single mode mathematical model of the beam using a Runge-Kutta algorithm is shown to give results qualitatively similar to experimental observations.     

Optimal Shape of Column with Own Weight: Bi and Single Modal Optimization

The optimality conditions, via Pontryagin’s maximum principle, in the case of bimodal optimization of columns are derived. When these conditions are applied to the stability of a compressed column with own weight, the problem of determining the optimal cross–sectional area function is reduced to the solution of a nonlinear boundary value problem. Two specific problems are analyzed in detail. In Problem 1, that is new, the shape of a heavy compressed column with clamped ends stable against buckling and having minimal volume is determined. In Problem 2, formulated by Keller and Niordson, optimal shape of a vertical column with one end clamped the other end free is determined.     

Surface and non-local effects for non-linear analysis of Timoshenko beams

In this paper, we present a non-local non-linear finite element formulation for the Timoshenko beam theory. The proposed formulation also takes into consideration the surface stress effects. Eringen׳s non-local differential model has been used to rewrite the non-local stress resultants in terms of non-local displacements. Geometric non-linearities are taken into account by using the Green–Lagrange strain tensor. A C⁰ beam element with three degrees of freedom has been developed. Numerical solutions are obtained by performing a non-linear analysis for bending and free vibration cases. Simply supported and clamped boundary conditions have been considered in the numerical examples. A parametric study has been performed to understand the effect of non-local parameter and surface stresses on deflection and vibration characteristics of the beam. The solutions are compared with the analytical solutions available in the literature. It has been shown that non-local effect does not exist in the nano-cantilever beam (Euler–Bernoulli beam) subjected to concentrated load at the end. However, there is a significant effect of non-local parameter on deflections for other load cases such as uniformly distributed load and sinusoidally distributed load (Cheng et al. (2015) [10]). In this work it has been shown that for a cantilever beam with concentrated load at free end, there is definitely a dependency on non-local parameter when Timoshenko beam theory is used. Also the effect of local and non-local boundary conditions has been demonstrated in this example. The example has also been worked out for other loading cases such as uniformly distributed force and sinusoidally varying force. The effect of the local or non-local boundary conditions on the end deflection in all these cases has also been brought out.     

Twelve shear surface waves guided by clamped/free boundaries in magneto-electro-elastic materials

It is shown that surface waves with 12 different velocities in the cases of different magneto-electrical boundary conditions can be guided by the interface of two identical magneto-electro-elastic half-spaces. The plane boundary of one of the half-spaces is clamped while the plane boundary of the other one is free of stresses. The 12 velocities of propagation of these surface waves are obtained is explicit forms. It is shown that the number of different surface wave velocities decreases from 12 to 2 if the magneto-electro-elastic material is changed to a piezoelectric material.     

Optimal shape and non-homogeneity of a non-uniformly compressed column

The best possible distribution of Young's modulus and/or the cross-sectional area is found for a column which, for a given volume and length, carries the maximum possible axial loads which are non-uniformly distributed along its length and concentrated at the end-points. The column is elastically clamped at one end and free at the other, where the concentrated axial load is applied. The design variables are subject to upper and lower bounds. Sufficient optimality conditions are derived for a given function to be a solution of the optimization problem. The procedure to determine the optimal solutions is described. Numerical results are obtained by employing an iterative computational technique.     

Full-wave numerical simulation of nonlinear dissipative acoustic standing waves in wind instruments

A finite volume full-wave method is used to simulate nonlinear dissipative acoustic propagation in ducts with a circular cross-section. Thermoviscous dissipative effects, due to bulk viscosity and shear viscosity in the boundary layer adjacent to the duct walls, are also considered. The propagation is assumed to be axisymmetric, and two different geometries are considered: a straight cylindrical tube, and a cylindrical tube joined smoothly to a slowly-flaring bell. Of special interest is the study of the onset of standing waves in the nonlinear regime. The full-wave numerical scheme is particularly well-adapted for this purpose, as it is not necessary to impose boundary conditions at the open end of the duct. A simplified model of excitation is adopted, where the lips are replaced by a spring–mass system which behaves like a pressure valve with a single degree of freedom. The full system behaves as expected, with a feedback cycle established between the pressure valve and the air column. The simulation is validated successfully in the linear regime using a theoretical solution. It is shown that increasing the stiffness of the lips leads to discrete jumps in playing frequency, which is behaviour typical of brass instruments. In the nonlinear regime, shock formation is observed for sufficiently high amplitudes of oscillation, and the radiation of these shock waves by the open end of the ducts can be visualised in the time-domain, along with edge-diffraction effects. The formation and evolution of standing waves in the nonlinear regime, where the effect of these shocks is very noticeable, is also examined.     

Postbuckling behavior of variable-arc-length elastica connected with a rotational spring joint including the effect of configurational force

This paper aims to study the behavior of a variable-arc-length (VAL) elastica subjected to the end loading, where a rotational spring joint is placed within the span length of the elastica. One end of the elastica is rested on the pinned support while the other end is placed into the sleeve support. The length of the elastica can be fed into the system through sleeve support by the end thrust where the effect of configurational force has been considered. A rotational spring joint is located within the span length of the elastica. From the equilibrium equations, moment–curvature expression, geometric relations, and boundary conditions, the closed-form solution in terms of elliptic integral of the first and second kinds can be demonstrated. The results obtained from elliptic integral method are validated with those from the shooting method and they are in excellent agreement. In order to interpret the behavior of the elastica, load–deflection curves and equilibrium shapes are established. Interesting features of the results are demonstrated. Particularly, when the stiffness of the spring joint becomes zero, the secondary buckling and the multiple equilibrium shapes can be captured in which the stable equilibrium shapes can be evaluated by using the vibration analysis. For a low value of the stiffness of the spring joint, the elastica has a possibility to exhibit the hardening behavior. When the stiffness of the spring joint becomes large, the elastica shows the softening behavior and its shape is identical to a single portion of VAL elastica.     

Critical velocities in fluid-conveying single-walled carbon nanotubes embedded in an elastic foundation

The problem of stability of fluid-conveying carbon nanotubes embedded in an elastic medium is investigated in this paper. A nonlocal continuum mechanics formulation, which takes the small length scale effects into consideration, is utilized to derive the governing fourth-order partial differential equations. The Fourier series method is used for the case of the pinned–pinned boundary condition of the tube. The Galerkin technique is utilized to find a solution of the governing equation for the case of the clamped–clamped boundary. Closed-form expressions for the critical flow velocity are obtained for different values of the Winkler and Pasternak foundation stiffness parameters. Moreover, new and interesting results are also reported for varying values of the nonlocal length parameter. It is observed that the nonlocal length parameter along with the Winkler and Pasternak foundation stiffness parameters exert considerable effects on the critical velocities of the fluid flow in nanotubes.     

DYNAMICS AND STABILITY OF PINNED–CLAMPED AND CLAMPED–PINNED CYLINDRICAL SHELLS CONVEYING FLUID

The paper examines the dynamics and stability of fluid-conveying cylindrical shells having pinned–clamped or clamped–pinned boundary conditions, where “pinned” is an abbreviation for “simply supported”. Flügge's equations are used to describe the shell motion, while the fluid-dynamic perturbation pressure is obtained utilizing the linearized potential flow theory. The solution is obtained using two methods — the travelling wave method and the Fourier-transform approach. The results obtained by both methods suggest that the negative damping of the clamped–pinned systems and positive damping of the pinned–clamped systems, observed by previous investigators for any arbitrarily small flow velocity, are simply numerical artefacts; this is reinforced by energy considerations, in which the work done by the fluid on the shell is shown to be zero. Hence, it is concluded that both systems are conservative.     

Optimal shape of an elastic column on elastic foundation

By using Pontryagin's maximum principle we determine the shape of an elastic compressed column on elastic, Winkler type foundation. We assume that the column has clamped ends. The optimality conditions for the case of bimodal optimization are derived. It is shown that the optimal cross-sectional area function is determined from the solution of a nonlinear boundary value problem. In the special case of a compressed column with no foundation, the optimality condition and the solution obtained earlier are recovered.     

Non-linear in-plane analysis and buckling of pinned–fixed shallow arches subjected to a central concentrated load

This paper is concerned with an analytical study of the non-linear elastic in-plane behaviour and buckling of pinned–fixed shallow circular arches that are subjected to a central concentrated radial load. Because the boundary conditions provided by the pinned support and fixed support of a pinned–fixed arch are quite different from those of a pinned–pinned or a fixed–fixed arch, the non-linear behaviour of a pinned–fixed arch is more complicated than that of its pinned–pinned or fixed–fixed counterpart. Analytical solutions for the non-linear equilibrium path for shallow pinned–fixed circular arches are derived. The non-linear equilibrium path for a pinned–fixed arch may have one or three unstable equilibrium paths and may include two or four limit points. This is different from pinned–pinned and fixed–fixed arches that have only two limit points. The number of limit points in the non-linear equilibrium path of a pinned–fixed arch depends on the slenderness and the included angle of the arch. The switches in terms of an arch geometry parameter, which is introduced in the paper, are derived for distinguishing between arches with two limit points and those with four limit points and for distinguishing between a pinned–fixed arch and a beam curved in-elevation. It is also shown that a pinned–fixed arch under a central concentrated load can buckle in a limit point mode, but cannot buckle in a bifurcation mode. This contrasts with the buckling behaviour of pinned–pinned or fixed–fixed arches under a central concentrated load, which may buckle both in a bifurcation mode and in a limit point mode. An analytical solution for the limit point buckling load of shallow pinned–fixed circular arches is also derived. Comparisons with finite element results show that the analytical solutions can accurately predict the non-linear buckling and postbuckling behaviour of shallow pinned–fixed arches. Although the solutions are derived for shallow pinned–fixed arches, comparisons with the finite element results demonstrate that they can also provide reasonable predictions for the buckling load of deep pinned–fixed arches under a central concentrated load.     

Buckling optimization of flexible columns

A novel approach to the optimization of flexible columns against buckling is presented. Previous published studies, considering either continuous or discrete finite element models, are always constrained to specific relations between stiffness and mass distributions of the column. These, besides yielding impractical configurations that do not conform to manufacturing and production requirements, result in designs that are certainly suboptimal. The present model formulation considers columns that can be practically made of uniform segments with the true design variables defined to be the cross-sectional area, radius of gyration and length of each segment. Exact structural analysis is performed, ensuring the attainment of the absolute maximum critical buckling load for any number of segments, type of cross section and type of boundary conditions. Detailed results are presented and discussed for clamped columns having either solid or tubular cross-sectional configurations, where useful design trends have been recommended for optimum patterns with two, three and more segments. It is shown that the developed optimization model, which is not restricted to specific properties of the cross section, can give higher values of the critical load than those obtained from constrained-continuous shape optimization. In fact, the model has succeeded in arriving at the global optimal column designs having the absolute maximum buckling load without violating the economic feasibility requirements.     

Refined theories may be needed for vibration analysis of structures with overhang

The effect of shear deformation and rotary inertia terms on the free vibration of a beam with overhang was investigated. A recently proposed modified Timoshenko-type equations of motion were used to analyze the vibration of the structure. Two different sets of boundary conditions, with either a fixed or hinged end support, were studied. The results were compared with those obtained for the classical Bernoulli–Euler beam theory. The comparison shows that for a hinged end beam with very long overhang, where the span between the supports is less than one tenth of the overall beam length, the classical theory significantly overestimates the values of the fundamental natural frequencies, even for isotropic shear rigidity. On the other hand, the span effect is reversed for the clamped end beam, for which a relatively significant difference between the classical theory and shear theory results may occur only for a long span. For transversely isotropic beams, the refined theory predictions of the fundamental natural frequencies may be much smaller than those obtained through the rigid shear theory, especially for short span hinged end beams and long span clamped end beams.     

On constructing a Green’s function for a semi-infinite beam with boundary damping

The main aim of this paper is to contribute to the construction of Green’s functions for initial boundary value problems for fourth order partial differential equations. In this paper, we consider a transversely vibrating homogeneous semi-infinite beam with classical boundary conditions such as pinned, sliding, clamped or with a non-classical boundary conditions such as dampers. This problem is of important interest in the context of the foundation of exact solutions for semi-infinite beams with boundary damping. The Green’s functions are explicitly given by using the method of Laplace transforms. The analytical results are validated by references and numerical methods. It is shown how the general solution for a semi-infinite beam equation with boundary damping can be constructed by the Green’s function method, and how damping properties can be obtained.     

Bucklewaves

Motivated by a selection of results on the plastic buckling of column members within a sandwich plate core where one face of the sandwich is subject to an intense impulse, the problem addressed is one where lateral buckling takes place simultaneously as a compressive axial wave propagates down the member. The bucklewave problem is modeled as an infinitely long column (or wide plate) which is clamped against lateral deflection at the end where velocity is imposed and has a moving clamped condition coinciding with the front of the plastic compression wave. The model reveals that a column or plate suddenly compressed into the plastic range is dynamically stabilized against lateral buckling for lengths that are significantly longer than the corresponding length at which the member would buckle quasi-statically. This stabilization has significant implications for energy absorption under intense dynamic loading. The analysis method is benchmarked against a simpler, but mathematically analogous problem, for which closed form solutions are available: the dynamics of a guitar string lengthening at constant velocity.     

Enhancement of elementary beam theories in order to obtain exact solutions for elastic rectangular beams

Boley's method is utilized in order to show that the elementary Bernoulli–Euler beam theory can be enhanced such that exact solutions of the plane-stress theory of linear elasticity are obtained for force loaded rectangular beams. An equivalent enhancement is derived for the elementary Timoshenko theory of beams. The enhancement terms act analogous to thermal loadings; they follow from the force loading of the rectangular beam in an explicit form. The resulting boundary value problem of fourth order can be efficiently solved by means of symbolic computer codes. As an illustrative example, a redundant beam is studied, which is simply supported at one end, and which is clamped at the other end. Outcomes for three alternative clamped end boundary conditions are compared.     

Exact solution of the multi-cracked Euler–Bernoulli column

The use of distributions (generalized functions) is a powerful tool to treat singularities in structural mechanics and, besides providing a mathematical modelling, their capability of leading to closed form exact solutions is shown in this paper. In particular, the problem of stability of the uniform Euler–Bernoulli column in presence of multiple concentrated cracks, subjected to an axial compression load, under general boundary conditions is tackled. Concentrated cracks are modelled by means of Dirac’s delta distributions. An integration procedure of the fourth order differential governing equation, which is not allowed by the classical distribution theory, is proposed. The exact buckling mode solution of the column, as functions of four integration constants, and the corresponding exact buckling load equation for any number, position and intensity of the cracks are presented. As an example a parametric study of the multi-cracked simply supported and clamped–clamped Euler–Bernoulli columns is presented.     

Enhancing flutter and buckling capacity of column by piezoelectric layers

This paper illustrates the use of a pair of piezoelectric layers in increasing the flutter and buckling capacity of a column subjected to a follower force. The column is fixed at one end while the other one is free to rotate but constrained transversely by a spring. The mathematical formulation is presented and solved numerically. The effect of the spring stiffness on the capacity and type of instability of the column is first illustrated numerically for the case without any piezoelectric actuators. A transition value for the stiffness can be identified, below which the column fails by flutter and above which the column buckles. Next, an external voltage is applied on the piezoelectric layers bonded on the surfaces of the column, which induces locally a pair of tensile follower force. This has the effect of increasing the capacity of the column as the voltage increases while the transition stiffness remains virtually unchanged for a given size and location of piezoelectric actuators. It is also shown that the capacity of the column increases with longer layers for a fixed voltage. However, the location of the layers along the column determines the transition stiffness and hence has an effect on the type of failure for a fixed spring constant. Positioning towards the fixed end increases the flutter capacity whereas positioning away will result in an increase in buckling capacity.     

Dynamics of a pipe conveying fluid flexibly restrained at the ends

The subject of this paper is the study of dynamics and stability of a pipe flexibly supported at its ends and conveying fluid. First, the equation of motion of the system is derived via the extended form of Hamilton׳s principle for open systems. In the derivation, the effect of flexible supports, modelled as linear translational and rotational springs, is appropriately considered in the equation of motion rather than in the boundary conditions. The resulting equation of motion is then discretized via the Galerkin method in which the eigenfunctions of a free-free Euler–Bernoulli beam are utilized. Thus, a general set of second-order ordinary differential equations emerges, in which, by setting the stiffness of the end-springs suitably, one can readily investigate the dynamics of various systems with either classical or non-classical boundary conditions. Several numerical analyses are initially performed, in which the eigenvalues of a simplified system (a beam) with flexible end-supports are obtained and then compared with numerical results, so as to verify the equation of motion, in its simplified form. Then, the dynamics of a pinned-free pipe conveying fluid is systematically investigated, in which it is found that a pinned-free pipe conveying fluid is generally neutrally stable until it becomes unstable via a Hopf bifurcation leading to flutter. The next part of the paper is devoted to studying the dynamics of a pinned-free pipe additionally constrained at the pinned end by a rotational spring. A wide range of dynamical behaviour is seen as the mass ratio of the system (mass of the fluid to the fluid+pipe mass) varies. It is surprising to see that for a range of rotational spring stiffness, an increase in the stiffness makes the pipe less stable. Finally, a pipe conveying fluid supported only by a translational and a rotational spring at the upstream end is considered. For this system, the critical flow velocity is determined for various values of spring constants, and several Argand diagrams along with modal shapes of the unstable modes are presented. The dynamics of this system is found to be very complex and often unpredictable (unexpected).     

Transient response of laminated plates with arbitrary laminations and boundary conditions under general dynamic loadings

The dynamic analysis of laminated plates with various loading and boundary conditions is presented employing generalized differential quadrature (GDQ) method. The first-order shear deformation theory is considered to model the transient response of the plate. The GDQ technique together with Newmark integration scheme is employed to solve the system of transient equations governing dynamics of the plate. Different symmetric and asymmetric lamination sequences together with various combinations of clamped, simply supported, and free boundary conditions are considered. Particular interest of this study regards to asymmetric orthotropic plates having free edge and mixed boundary conditions. It is shown that the method provides reasonably accurate results with relatively small number of grid points. Comparison of the results with those of other methods demonstrates a very good agreement. It is also revealed that the present method offers similar order of accuracy for all variables including displacements and stress resultants.