Dynamics of a Flexible Cantilever Beam Carrying a Moving Mass

The motion of a flexible cantilever beam carrying a moving spring-mass system is investigated. The beam is assumed to be an Euler–Bernouli beam. The motion of the system is described by a set of two nonlinear coupled partial differential equations where the coupling terms have to be evaluated at the position of the mass. The nonlinearities arise due to the coupling between the mass and the beam. Due to the nonlinearities the system exhibits internal resonance which is investigated in this work. The equations of motion are solved numerically using the Rayleigh–Ritz method and an automatic ODE solver. An approximate solution using the perturbation method of multiple scales is also obtained.     

Non-linear longitudinal vibrations of non-slender piezoceramic rods

Characteristic non-linear effects can be observed, when piezoceramics are excited using weak electric fields. In experiments with longitudinal vibrations of piezoceramic rods, the behavior of a softening Duffing-oscillator including jump phenomena and multiple stable amplitude responses at the same excitation frequency and voltage is observed. Another phenomenon is the decrease of normalized amplitude responses with increasing excitation voltages. For such small stresses and weak electric fields as applied in the experiments, piezoceramics are usually described by linear constitutive equations around an operating point in the butterfly hysteresis curve. The non-linear effects under consideration were, e.g. observed and described by Beige and Schmidt [1,2], who investigated longitudinal plate vibrations using the piezoelectric 31-effect. They modeled these non-linearities using higher order quadratic and cubic elastic and electric terms. Typical non-linear effects, e.g. dependence of the resonance frequency on the amplitude, superharmonics in spectra and a non-linear relation between excitation voltage and vibration amplitude were also observed e.g. by von Wagner et al. [3] in piezo-beam systems. In the present paper, the work is extended to longitudinal vibrations of non-slender piezoceramic rods using the piezoelectric 33-effect. The non-linearities are modeled using an extended electric enthalpy density including non-linear quadratic and cubic elastic terms, coupling terms and electric terms. The equations of motion for the system under consideration are derived via the Ritz method using Hamilton's principle. An extended kinetic energy taking into consideration the transverse velocity is used to model the non-slender rods. The equations of motion are solved using perturbation techniques. In a second step, additional dissipative linear and non-linear terms are used in the model. The non-linear effects described in this paper may have strong influence on the relation between excitation voltage and response amplitude whenever piezoceramic actuators and structures are excited at resonance.     

Rayleigh–Ritz analysis for localized buckling of a strut on a softening foundation by Hermite functions

This paper proposes a Rayleigh–Ritz procedure for localized buckling of a strut on a non-linear elastic foundation. Firstly, the deflected shape of a strut is expanded into a series of Hermite orthogonal functions, which are proved energy-integrable in an infinite region. Secondly, the errors of the numerical integrations of Hermite functions on the infinite region are investigated and the suitable integral limit is proposed. Through the numerical investigation, it is demonstrated that the first thirty Hermite functions are usually enough to approximate the localized buckling pattern. The proposed method overcomes the disadvantages of the traditional methods, in which the trial functions in either Rayleigh–Ritz or Galerkin analysis are based on the perturbation analyses of the corresponding non-linear differential equation.     

Vibrations of a complex-shaped panel

The free vibrations of flexible shallow shells with complex planform are studied. To analyze the natural frequencies and modes of linear vibrations, the R-function and Rayleigh–Ritz methods are used. A discrete model is obtained using the Bubnov–Galerkin method. The nonlinear vibrations are studied by combining the nonlinear normal mode method and the multiple-scales method. Skeleton curves of natural vibrations are drawn     

An analytical formulation in 3D domain for the nonlinear response of piezoelectric slabs under weak electric fields

Piezoceramic materials exhibit different types of nonlinearities depending upon the magnitude of the mechanical and electric field strength within the body. Some of the nonlinear phenomena observed under weak electric fields near resonance frequency excitation are the presence of superharmonics in the response spectra and the jump phenomena etc. In this work, an analytical solution for the nonlinear response of rectangular piezoceramic slabs have been obtained by Rayleigh–Ritz method and perturbation technique in the 3-D domain using a generalized nonlinear electric enthalpy density function. Forced vibration experiments (excitation with electric field) have been conducted on a rectangular piezoceramic slab at varying electric field amplitudes and the analytical solutions have been shown to compare very well with the experimental results.     

Spacing effects on hydrodynamics of heave plates on offshore structures

The nonlinear viscous flow problem associated with a heaving vertical cylinder with two heave plates in the form of two circular disks attached is solved using a finite difference method. Numerical experiments are carried out to investigate the spanwise length effects on the hydrodynamic properties, such as added mass and damping coefficients. Over a Keulegan–Carpenter (KC) number range from 0.1 to 0.6 at frequency parameter (β=7.869×10⁷), calculations indicate that a KC-dependent critical value of spanwise length L/D_d (L—the distance between the disks, and D_d—the diameter of the disks) exists. A significant influence of L/D_d on the vortex shedding patterns around the disks and the hydrodynamic coefficients is revealed when L/D_d is smaller than the critical value due to strong interaction between vortices of different disks. Beyond that limit, however, both added mass and damping coefficients are shown to be rather stable, indicating the independence of the vortex shedding processes of different disks.     

Numerical results for the conservation of energy of Maxwell media for the Rayleigh–Stokes problem

This publication continues our studies of analytical solutions of the Rayleigh–Stokes problem for Maxwell fluids [J. Zierep, C. Fetecau, Energetic balance for the Rayleigh–Stokes problem of a Maxwell fluid, Int. J. Eng. Sci. 45 (2007) 617–627]. We start from the Fourier sine transform. The numerical result is given and discussed for the velocity u, the power of the wall shear stresses L, the dissipation Φand the boundary layer thickness δ. These new results are important for nature and technology.     

Nonlinear elastic analysis of planar curved beams loaded by co-planar concentrated forces

Summary In this paper an analytical procedure for the nonlinear elastic analysis of a cantilever planar curved beam, subjected to a concentrated co-planar force at its free end, is presented. According to this method the nonlinear differential equations describing the equilibrium of the deformed beam are decoupled and a solution in the form of elliptic integrals is obtained, in the case when the curvature of the initial beam is a linear function of the areS.

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Zusammenfassung In dieser Arbeit wird eine analytische Methode für die nichtlineare elastische Analyse einer freiträgigen ebener Tragbalkens, an desser freien Ende eine Konzentrierte Komplanare Kraft wirkt, entwickett. Nach dieser Methode die nichtlineare Differentialgleichungen, welche das Gleichgevicht des deformierten Tragbalkens beschreiben, abgekoppelt, und es wird eine geschlossene Lösung mit Hilfe elliptischer Integralen erreicht und im falle wo die Krümmung de gegebenen Tragbalkens eine lineare function des BogensS ist.

     

Piezomagnetic and piezoelectric poling effects on mode I and II crack initiation behavior of magnetoelectroelastic materials

The ferrite and ferroelectric phase of magnetoelectroelastic (MEE) material can be selected and processed to control the macroscopic behavior of electron devices using continuum mechanics models. Once macro- and/or microdefects appear, the highly intensified magnetic and electric energy localization could alter the response significantly to change the design performance. Alignment of poling directions of piezomagnetic and piezoelectric materials can add to the complexity of the MEE material behavior to which this study will be concerned with.Appropriate balance of distortional and dilatational energy density is no longer obvious when a material possesses anisotropy and/or nonhomogeneity. An excess of the former could result in unwanted geometric change while the latter may lead to unexpected fracture initiation. Such information can be evaluated quantitatively from the stationary values of the energy density function dW/dV. The maxima and minima have been known to coincide, respectively, with possible locations of permanent shape change and crack initiation regardless of material and loading type. The direction of poling with respect to a line crack and the material microstructure described by the constitutive coefficients will be specified explicitly with reference to the applied magnetic field, electric field and mechanical stress, both normal and shear. The crack initiation load and direction could be predicted by finding the direction for which the volume change is the largest. In contrast to intuition, change in poling directions can influence the cracking behavior of MEE dramatically. This will be demonstrated by the numerical results for the BaTiO₃–CoFe₂O₄ composite having different volume fractions where BaTiO₃ and CoFe₂O₄ are, respectively, the inclusion and matrix.To be emphasized is that mode I and II crack behavior will not have the same definition as that in classical fracture mechanics where load and crack extension symmetry would coincide. A striking result is found for a mode II crack. By keeping the magnetic poling fixed, a reversal of electric poling changed the crack initiation angle from θ₀=+80° to θ₀=−80° using the line extending ahead of the crack as the reference. This effect is also sensitive to the distance from the crack tip. Displayed and discussed are results for r/a=10⁻⁴ and 10⁻¹. Because the theory of magnetoelectroelasticity used in the analysis is based on the assumption of equilibrium where the influence of material microstructure is homogenized, the local space and temporal effects must be interpreted accordingly. Among them are the maximum values of (dW/dV)_max and (dW/dV)_min which refer to as possible sites of yielding and fracture. Since time and size are homogenized, it is implicitly understood that there is more time for yielding as compared to fracture being a more sudden process. This renders a higher dW/dV in contrast to that for fracture. Put it differently, a lower dW/dV with a shorter time for release could be more detrimental.     

Global Bifurcations and Chaotic Dynamics in Nonlinear Nonplanar Oscillations of a Parametrically Excited Cantilever Beam

This paper presents the analysis of the global bifurcations and chaotic dynamics for the nonlinear nonplanar oscillations of a cantilever beam subjected to a harmonic axial excitation and transverse excitations at the free end. The governing nonlinear equations of nonplanar motion with parametric and external excitations are obtained. The Galerkin procedure is applied to the partial differential governing equation to obtain a two-degree-of-freedom nonlinear system with parametric and forcing excitations. The resonant case considered here is 2:1 internal resonance, principal parametric resonance-1/2 subharmonic resonance for the in-plane mode and fundamental parametric resonance–primary resonance for the out-of-plane mode. The parametrically and externally excited system is transformed to the averaged equations by using the method of multiple scales. From the averaged equation obtained here, the theory of normal form is applied to find the explicit formulas of normal forms associated with a double zero and a pair of pure imaginary eigenvalues. Based on the normal form obtained above, a global perturbation method is utilized to analyze the global bifurcations and chaotic dynamics in the nonlinear nonplanar oscillations of the cantilever beam. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Silnikov type single-pulse homoclinic orbit in the averaged equation for the nonlinear nonplanar oscillations of the cantilever beam. These results show that the chaotic motions can occur in the nonlinear nonplanar oscillations of the cantilever beam. Numerical simulations verify the analytical predictions.     

Mechanisms of brittle-ductile transition in toughened thermoplastics

The objective of this work was to investigate the mechanism of brittle-ductile transition in toughened polymers. Two systems, namely, a rubber-toughened nylon 66 (Zytel ST-801) and a high impact polystyrene (HIPS), were chosen for this study. The samples were prepared by injection molding and were tested in three-point bending under various loading rates and temperatures. The brittle-ductile transition temperature (T_b–d) was determined from the observed fracture behavior as a function of temperature. Molecular relaxation temperatures of the polymers were measured by mechanical spectroscopy at various frequencies. The correlation between temperature and loading rate was estimated using the Arrhenius equation. The results show that T_b–d of Zytel ST-801 is only slightly affected by the loading rate, whereas T_b–d of HIPS strongly increases with deformation rate. It is found that for the former, within the experimental errors, an increase in T_b–d with loading rate corresponds to the shift in the secondary relaxation temperature T_b of the nylon 66 matrix. For the latter however, the increase in T_b–d is related to the glass/rubber relaxation of the polystyrene matrix. It seems that the type of molecular relaxation controlling the brittle-ductile transition corresponds to that with lower activation energy.     

Influence of heat transfer on the dynamic response of a spherical gas/vapour bubble

The standard approach to analyse the bubble motion is the well known Rayleigh–Plesset equation. When applying the toolbox of nonlinear dynamical systems to this problem several aspects of physical modelling are usually sacrificed. Particularly in vapour bubbles the heat transfer in the liquid domain has a significant effect on the bubble motion; therefore the nonlinear energy equation coupled with the Rayleigh–Plesset equation must be solved. The main aim of this paper is to find an efficient numerical method to transform the energy equation into an ODE system, which, after coupling with the Rayleigh–Plesset equation can be analysed with the help of bifurcation theory. Due to the strong nonlinearity and violent bubble motions the computational effort can be high, thus it is essential to reduce the size of the problem as much as possible. In the first part of the paper finite difference, Galerkin and spectral collocation methods are examined and compared in terms of efficiency. In the second part free and forced oscillations are analysed with an emphasis on the influence of heat transfer. In the case of forced oscillations the unstable branches of the amplification diagrams are also computed.     

Modelling the Stationary Vibrations and Dissipative Heating of Thin-Walled Inelastic Elements with Piezoactive Layers

A coupled dynamic problem of thermoelectromechanics for thin-walled multilayer elements is formulated based on a geometrically nonlinear theory and the Kirchhoff–Love hypotheses. In the case of harmonic loading, an approximate formulation is given using the concept of complex moduli to characterize the cyclic properties of the material. The model problem on forced vibrations of sandwich beam, whose core layer is made of a passive physically nonlinear material, and face layers, of a viscoelastic piezoactive material, is considered as an example to demonstrate the possibility of damping the vibrations by applying harmonic voltage to the oppositely polarized layers of the beam. Substantiation is given for a linear control law with a complex coefficient for the electric potential, which provides damping of vibrations in the first symmetric mode at the linear and nonlinear stages of deformation. The stress–strain state and dissipative-heating temperature are studied     

Solutions of Weakly Nonlinear Systems of Ordinary Differential Equations Bounded on R

We obtain conditions for the existence of solutions bounded on the entire axis R for weakly nonlinear systems of ordinary differential equations in the case where the corresponding unperturbed homogeneous linear differential system is exponentially dichotomous on the semiaxes R ₊ and R _–.     

Order Reduction of Parametrically Excited Nonlinear Systems: Techniques and Applications

The basic problem of order reduction of nonlinear systems with time periodic coefficients is considered in state space and in direct second order (structural) form. In state space order reduction methods, the equations of motion are expressed as a set of first order equations and transformed using the Lyapunov–Floquet (L–F) transformation such that the linear parts of new set of equations are time invariant. At this stage, four order reduction methodologies, namely linear, nonlinear projection via singular perturbation, post-processing approach and invariant manifold technique, are suggested. The invariant manifold technique yields a unique ‘reducibility condition’ that provides the conditions under which an accurate nonlinear order reduction is possible. Unlike perturbation or averaging type approaches, the parametric excitation term is not assumed to be small. An alternate approach of deriving reduced order models in direct second order form is also presented. Here the system is converted into an equivalent second order nonlinear system with time invariant linear system matrices and periodically modulated nonlinearities via the L–F and other canonical transformations. Then a master-slave separation of degrees of freedom is used and a nonlinear relation between the slave coordinates and the master coordinates is constructed. This method yields the same ‘reducibility conditions’ obtained by invariant manifold approach in state space. Some examples are given to show potential applications to real problems using above mentioned methodologies. Order reduction possibilities and results for various cases including ‘parametric’, ‘internal’, ‘true internal’ and ‘true combination resonances’ are discussed. A generalization of these ideas to periodic-quasiperiodic systems is included and demonstrated by means of an example.     

Ein Beitrag zur Kinematik der Kontinua,Teil I

Übersicht Die Deformationsmatrix D = (x ^k , k) = (d _ik ) eines isotropen Kontinuums hat im zweidimensionalen Fall Elemente als Orthogonalfaktor, die nur noch von der Invarianten (d ₁₂–d ₂₁)/(d ₁₁ + d ₂₂) abhängen. Wählt man eine zweifach stetig differenzierbare Funktion dieser Invarianten als volumenbezogene Spannungsenergie, dann ergeben die Bewegungsgleichungen Normalflächen, die aus zwei Kugelflächen mit variablem Radius bestehen. Die Strahlen werden dabei normal zur Wellenoberfläche, wie dies in der Theorie linearer Kontinua der Fall ist.

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Summary In the two-dimensional case, the deformation matrix D = (x ^k , k) = (d _ik ) of an isotropic continuum has elements as orthogonal factors which depend only on the invariant (d ₁₂–d ₂₁)/(d ₁₁ + d ₂₂). If an arbitrary twice continuously differentiable function of this invariant is chosen as strain energy per unit volume, the equations of motion give normal surfaces which separate into two spherical surfaces with variable radius. In this way the rays become normal to the wave-surface, as is the case in linear continuum theory.

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Herrn Prof. Dr.-Ing. E. Pestel zum 60. Geburtstag gewidmet.     

Direct Numerical Simulation of a Gaussian acoustic wave interaction with a turbulent premixed flame

The interaction of a Gaussian negative pulse with a H₂/O₂/N₂ turbulent premixed flame is examined using Direct Numerical Simulation (DNS). Transport properties and chemical kinetics are described in a very detailed manner. An extended nonlinear local Rayleigh's criterion, for laminar as well as turbulent, premixed or nonpremixed flames, is proposed. Situations in which amplification or attenuation occur are listed. Calculations of a turbulent flame are then carried out with and without an acoustic wave and results are recorded at the same time. The influence of acoustic wave/turbulent flame interaction is obtained by a simple difference. It is shown that longitudinal and transverse velocity components are perturbed by the turbulent flame. Moreover, the vorticity induced by the acoustic wave is observed to be weak. Finally, Rayleigh's criterion shows that wave amplification occurs punctually. To cite this article: A. Laverdant, D. Thévenin, C. R. Mecanique 333 (2005).     

Nonlinear vibration analysis of isotropic cantilever plate with viscoelastic laminate

The nonlinear vibration of an isotropic cantilever plate with viscoelastic laminate is investigated in this article. Based on the Von Karman’s nonlinear geometry and using the methods of multiple scales and finite difference, the dimensionless nonlinear equations of motion are analyzed and solved. The solvability condition of nonlinear equations is obtained by eliminating secular terms and, finally, nonlinear natural frequencies and mode-shapes are obtained. Knowing that the linear vibration of this type of plate does not have exact solution, Ritz method is employed to obtain semi-analytical nonlinear mode-shapes of transverse vibration of this plate. Airy stress function and Galerkin method are employed to reduce nonlinear PDEs into an ODE of duffing type. Stability of plate and chaotic behavior are investigated by Runge–Kutta method. Poincare section diagrams are in good agreement with results of Lyapunov criteria.     

Radial Vibrations and Heating of a Ring Piezoplate under Electric Excitation Applied to Nonuniformly Electroded Surfaces

Radial vibrations and dissipative heating of a polarized piezoceramic ring plate are studied. The plate is excited by a harmonic electric field applied to nonuniformly electroded surfaces of the plate. The viscoelastic behavior of piezoceramics is described in terms of complex quantities. An analytical solution is found in the case of quasistatic harmonic loading. The dynamic nonlinear problem of coupled thermoviscoelasticity is solved with regard for the temperature dependence of the properties of piezoceramics by step-by-step integration in time, using the numerical methods of discrete orthogonalization and finite differences. A numerical analysis is conducted for TsTStBS-2 piezoceramics to study the influence of partial electroding on the stress–strain distribution, natural frequency, and amplitude–frequency and temperature–frequency characteristics