Showing papers in "Acta Mechanica in 2003"

Elastoviscoplastic constitutive frameworks for generalized continua

Abstract: A unifying thermomechanical constitutive framework for generalized continua including additional degrees of freedom or/and the second gradient of displacement is presented. Based on the analysis of the dissipation, state laws, flow rules and evolution equations are proposed for Cosserat, strain gradient and micromorphic continua. The case of the gradient of internal variable approach is also incorporated by regarding the nonlocal internal variable as an actual additional degree of freedom. The consistency of the continuum thermodynamical framework is ensured by the introduction of a viscoplastic pseudo–potential of dissipation, thus extending the classical class of so–called standard material models to generalized continua.

Peristaltic pumping of a micropolar fluid in a tube

Abstract: Peristaltic transport of a micropolar fluid in a circular tube is studied under low Reynolds number and long wavelength approximations. The closed form solutions are obtained for velocity, microrotation components, as well as the stream function and they contain new additional parameters namely, N the coupling number and m the micropolar parameter. In the case of free pumping (pressure difference Δp=0) the difference in pumping flux is observed to be very small for Newtonian and micropolar fluids but in the case of pumping (Δp>0) the characteristics are significantly altered for different N and m. It is observed that the peristalsis in micropolar fluids works as a pump against a greater pressure rise compared with a Newtonian fluid. Streamline patterns which depict trapping phenomena are presented for different parameter ranges. The limit on the trapping of the center streamline is obtained. The effects of N and m on friction force for different Δp are discussed.

A Comparative Study of Biot’s Theory and the Linear Theory of Porous Media for Wave Propagation Problems

Abstract: Wave propagation in porous media is an important topic for example in geomechanics or oil-industry. Especially due to the interplay of the solid skeleton with the fluid the so-called second compressional wave appears. The existence of this wave is reported in the literature not only for Biot's theory (BT) but also for theoretical approaches based on the Theory of Porous Media (TPM – mixture theory extended by the concept of volume fractions). Assuming a geometrically linear description (small displacements and small deformation gradients) and linear constitutive equations (Hooke's law) the governing equations are derived for both theories, BT and the TPM, respectively. In both cases, the solid displacements and the pore pressure are the primary unknowns. Note that this is only possible in the Laplace domain leading to the same structure of the coupled differential equations for both approaches. But the differential equations arising in BT and TPM possess different coefficients with different physical interpretations. Correlating these coefficients to each other leads to the well-known problem of Biot's “apparent mass density”. Furthermore, some inconsistencies are observed if Biot's stress coefficient is correlated to the structure arising in TPM. In addition to the comparison of the governing equations and the identification of the model parameters, the displacement and pressure solutions of both theories are presented for a one-dimensional column. The results show good agreement between both approaches in case of incompressible constituents whereas in case of compressible constituents large differences appear.

Thermomechanical postbuckling analysis of functionally graded plates and shallow cylindrical shells

Abstract: In this paper, an analytic solution is provided for the postbuckling behavior of plates and shallow cylindrical shells made of functionally graded materials under edge compressive loads and a temperature field. The material properties of the functionally graded shells are assumed to vary continuously through the thickness of the shell according to a power law distribution of the volume fraction of the constituents. The fundamental equations for thin rectangular shallow shells of FGM are obtained using the von Karman theory for large transverse deflection, and the solution is obtained in terms of mixed Fourier series. The effect of material properties, boundary conditions and thermomechanical loading on the buckling behavior and stress field are determined and discussed. The results reveal that thermomechanical coupling effects and the boundary conditions play a major role in dictating the response of the functionally graded plates and shells under the action of edge compressive loads.

Dynamic buckling of functionally graded cylindrical thin shells under non-periodic impulsive loading

Abstract: In this paper, a formulation for the buckling of cylindrical thin shells made of functionally graded material (FGM) composed of ceramic and metal subjected to external pressure varying as a power function of time is presented. The properties are graded in the thickness direction according to a volume fraction power-law distribution. The modified Donnell type dynamic stability and compatibility equations are obtained using Love’s shell theory. Applying Galerkin’s method and then applying a Ritz type variational method to these equations for different initial conditions and taking large values of the loading parameters into consideration, analytic solutions are obtained for critical parameter values. The results show that the critical parameters are affected by the configurations of the constituent materials, loading parameters variations and the power of time in the external pressure expression variations. Comparing the results with those in the literature validates the present analysis.

Application of the differential transformation method to the solutions of Falkner-Skan wedge flow

Abstract: This paper adopts the differential transformation method to investigate the velocity and shear stress fields associated with the Flakner-Skan boundary-layer problem. A group of transformations are used to reduce the boundary value problem into a pair of initial value problems, which are then solved by means of the differential transformation method. The proposed method yields closed series solutions of the boundary layer equations, which can then be calculated numerically. Numerical results for the dimensionless velocity and the shear stress profiles of the wedge flow are presented graphically for different values of β. It is found that the current results are in good agreement with those provided by other numerical methods. Therefore, the proposed method is proven to be an effective scheme for the solution of nonlinear boundary-layer problems.

Eshelby inclusion of arbitrary shape in an anisotropic plane or half-plane

Abstract: Analytical solution for Eshelby's problem of an anisotropic non-elliptical inclusion remains a challenging problem. In this paper, a simple method is presented to obtain an analytical solution for Eshelby's problem of an inclusion of arbitrary shape within an anisotropic plane or half-plane of the same elastic constants. The method is based on an observation that the interface conditions for arbitrary inclusion-shape can be written in a decoupled form in which three unknown Stroh's functions are decoupled from each other. The solution is given in terms of three auxiliary functions constructed by three conformal mappings which map the exteriors of three image curves of the inclusion boundary, defined by three Stroh's variables, onto the exterior of the unit circle. With aid of these auxiliary functions, techniques of analytical continuation can be applied to the inclusion of any shape. The solution is given in the physical plane rather than in the image plane, and is exact provided that the expansion of every mapping function includes only a finite number of terms. On the other hand, if at least one of the mapping functions includes infinite terms, a truncated polynomial mapping should be used, and thus the method gives an approximate solution. A remarkable feature of the present method is that it gives elementary expressions for the internal stress field within an inclusion in an anisotropic entire plane. Elliptical and polygonal inclusions are used to illustrate the construction of the auxiliary functions and the details of the method.

Vibration of turbomachinery rotating blades made-up of functionally graded materials and operating in a high temperature field

Abstract: The vibration of turbomachinery rotating blades made-up of functionally graded materials and operating in a temperature field is considered. In this context, the blade is modeled as a thin-walled beam mounted on a rigid hub at a presetting angle, rotating with a constant angular velocity, and exposed to a steady temperature field of a prescribed gradient through the blade wall thickness. The effect of the initial twist of the blade is also taken into consideration. Results are presented for two constituents, metal-ceramic based materials that are assumed to be temperature-dependent and graded in the thickness direction according to a simple power law distribution. Numerical results highlighting the effects of the volume fraction in conjunction with those of the presetting and pretwist angles, temperature gradient, rotating speed and hub radius are presented, and pertinent conclusions are outlined.

An Eshelbian approach to the nonlinear mechanics of constrained solid-fluid mixtures

Abstract: Looking at rational mixture theories within the context of a new perspective, this work aims to put forward a proposal for an Eshelbian approach to the nonlinear mechanics of a constrained solid-fluid mixture, made up of an inhomogeneous poroelastic solid and an inviscid compressible fluid, which do not undergo any chemical reaction.

Constructal multi-scale structure for maximal heat transfer density

Abstract: This paper presents a new concept for generating the multi-scale structure of a finite-size flow system that has maximum heat transfer density–maximum heat transfer rate installed in a fixed volume. Laminar forced convection and parallel isothermal blades fill the volume. The spacings between adjacent blades of progressively smaller scales are optimized based on constructal theory: the goal is maximum heat transfer density. The smaller blades are installed in the fresh-fluid regions that sandwich the tips of the boundary layers of longer blades. The overall pressure difference is constrained. As the number of length scales increases, the flow rate decreases and the volume averaged heat transfer density increases. There exists a smallest (cutoff) length scale below which heat transfer surfaces are no longer lined by distinct (slender) boundary layers. Multi-scale flow structures for maximum heat transfer rate density can be developed in an analogous fashion for natural convection. The constructal multi-scale algorithms are deduced from principles, unlike in fractal geometry where algorithms are assumed.

Similarity solutions for a moving-flat plate thermal boundary layer

Abstract: In this paper, the thermal boundary-layer problem of a semi-infinite flat plate moving in a constant velocity free stream is studied. The similarity equations with viscous dissipation for the thermal boundary-layer are derived and solved by numerical techniques. Under some specific conditions, the thermal boundary-layer similarity equation can be integrated analytically. The results are analyzed for very small Eckert number case and large Eckert number case. It is found that, for the two cases, wall heat fluxes will increase with the increase of the velocity ratio λ. With increasing Eckert number, the viscous dissipation heating will become dominant. However, for the Prandtl number when the Eckert number is small, it is found that wall heat fluxes will increase with increasing Prandtl number only for a certain range of velocity ratio λ. For the other range, the wall heat fluxes will have a maximum at a certain Prandtl number, and, when the Prandtl number is larger than the critical value, wall heat fluxes will decrease with increasing Prandtl number. Some examples of the lower solution branch are also presented to compare with the upper solution branch. It is found that the lower solution branch will result in lower heat fluxes at the wall.

Effect of time-periodic boundary temperatures/body force on Rayleigh-Benard convection in a ferromagnetic fluid

Abstract: We discuss the thermal instability in a layer of a ferromagnetic fluid when the boundaries of the layer are subjected to synchronous/asynchronous imposed time-periodic boundary temperatures (ITBT)/time–periodic body force (TBF). Only infinitesimal disturbances are considered. The Venezian approach is adopted in arriving at the critical Rayleigh and wave numbers for small amplitudes of ITBT. A perturbation solution in powers of the amplitude of the applied temperature field is obtained. When the ITBT at the two walls are synchronized then, for moderate frequency values, the role of magnetization in inducing sub-critical instabilities is delineated. A similar role is shown to be played by the Prandtl number. The magnetization parameters and Prandtl number have the opposite effect at large frequencies. The system is most stable when the ITBT is asynchronous. The effect of TBF on the onset of convection is found to be qualitatively similar to the effect of an asynchronous ITBT. Low Prandtl number fluids are shown to be more easily vulnerable to destabilization by TBF compared to very large Prandtl number fluids. The problem has relevance in many ferromagnetic fluid applications wherein regulation of thermal convection is called for.

Further study on a moving-wall boundary-layer problem with mass transfer

Abstract: The boundary-layer problem of a semi-infinite flat plate moving in a free stream with mass transfer is discussed in this paper. The paper extends the work of previous researchers to the general situations including mass injection as well as suction on the wall and the case of wall moving in the same direction as the free stream velocity. The analysis is concentrated on the wall drag. The solutions are obtained by numerical techniques. Under certain conditions, current results will reduce to those obtained by other researchers.

On spatial and material settings of thermo-hyperelastodynamics for open systems

Abstract: The present treatise aims at deriving a general framework for the thermodynamics of open systems typically encountered in chemo- or biomechanical applications. Due to the fact that an open system is allowed to constantly gain or lose mass, the classical conservation law of mass has to be recast into a balance equation balancing the rate of change of the current mass with a possible in- or outflux of matter and an additional volume source term. The influence of the generalized mass balance on the balance of momentum, kinetic energy, total energy and entropy is highlighted. Thereby, special focus is dedicated to the strict distinction between a volume specific and a mass specific format of the balance equations which is of no particular relevance in classical thermodynamics of closed systems. The change in density furnishes a typical example of a local rearrangement of material inhomogeneities which can be characterized most elegantly in the material setting. The set of balance equations for open systems will thus be derived for both, the spatial and the material motion problem. Thereby, we focus on the one hand on highlighting the remarkable duality between both approaches. On the other hand, special emphasis is placed on deriving appropriate relations between the spatial and the material motion quantities.

Elastomer bushing response: Experiments and finite element modeling

Abstract: Elastomer bushings are essential components in tuning suspension systems since they isolate vibration, reduce noise transmission, accommodate oscillatory motions and accept misalignment of axes. This work presents an experimental study in which bushings are subjected to radial, torsional and coupled radial-torsional modes of deformation. The experimental results show that the relationship between the forces and moments and their corresponding displacements and rotations is nonlinear and viscoelastic due to the nature of the elastomeric material. An interesting feature of the coupling response is that radial force decreases and then increases with torsion. The experimental results were used to assess bushing behavior and to determine the strength of radial-torsional coupling. The experimental results were also compared to finite element simulations of a model bushing. While finite element analysis predicted small displacements at the relaxed state reasonably well, the response to larger radial deformations and coupled deformations was not well captured.

On direct numerical treatment of hypersingular integral equations arising in mechanics and acoustics

Abstract: In this paper, we present a treatment of hypersingular integral equations, which have relevant applications in many problems of wave dynamics, elasticity and fluid mechanics with mixed boundary conditions. The main goal of the present work is the development of an efficient direct numerical collocation method. The paper also includes two examples taken from fracture mechanics and acoustics: a single crack in a linear isotropic elastic medium, and diffraction of a plane acoustic wave by a thin rigid screen.

Large deflection analysis of beams with variable stiffness

Abstract: In this paper, the Analog Equation Method (AEM), a BEM-based method, is employed to the nonlinear analysis of a Bernoulli-Euler beam with variable stiffness undergoing large deflections, under general boundary conditions which maybe nonlinear. As the cross-sectional properties of the beam vary along its axis, the coefficients of the differential equations governing the equilibrium of the beam are variable. The formulation is in terms of the displacements. The governing equations are derived in both deformed and undeformed configuration and the deviations of the two approaches are studied. Using the concept of the analog equation, the two coupled nonlinear differential equations with variable coefficients are replaced by two uncoupled linear ones pertaining to the axial and transverse deformation of a substitute beam with unit axial and bending stiffness, respectively, under fictitious load distributions. Besides the effectiveness and accuracy of the developed method, a significant advantage is that the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. Several beams are analyzed under various boundary conditions and loadings to illustrate the merits of the method as well as its applicability, efficiency and accuracy.

Boundary-layer similarity flows driven by a power-law shear over a permeable plane surface

Abstract: The boundary-layer similarity flow driven over a semi-infinite permeable flat plate by a power-law shear with asymptotic velocity profile U ∞(y)=βy α(y→∞,β>0) is considered in the presence of lateral suction or injection of the fluid (y denotes the coordinate normal to the plate). The analytically tractable cases α=−2/3 and α=−1/2 are examined in detail. It is shown that while for α=−2/3 the adjustment of the flow over an impermeable plate to the power-law shear is not possible, in the permeable cases the presence of suction allows for a family of boundary-layer solutions with the proper algebraic decay. The value of the skin friction corresponding to this family of solutions is given by the parameter s=9β3/(4f w ), where f w denotes the suction parameter. In the limiting case of a vanishing suction and a properly vanishing value of the parameter β (such that s=finite), this family of algebraically decaying solutions goes over into the (exponentially decaying) Glauert-jet. In the case α=−1/2, solutions showing the proper algebraic decay were found both for suction ( f w > 0) and injection ( f w <0) in the whole range −∞

Stress softening of elastomers in hydrostatic tension

Abstract: This study is concerned with inelastic effects of non-reinforcing carbon-black filled elastomers when subjected to periodic hydrostatic loading-unloading cycles in tension. During cyclic testing of sufficient magnitude, a critical state may be reached where microcavities suddenly grow inside the rubber, possibly initiated at sites of internal imperfections. As a result of cavitation damage the tensile bulk modulus in the natural configuration is reduced. A series of hydrostatic tension tests are performed at room temperature to provide new insight into the progressive deterioration of the bulk stiffness. We define dilatational stress softening as a phenomenon where the hydrostatic stress on unloading and subsequent submaximal reloading is significantly less than that on primary loading for the same volumetric strain. Dilatational stress softening during initial loading cycles and the permanent volumetric change upon unloading are not accounted for when the mechanical properties are represented in terms of a strain-energy function, i.e. if the material is modelled as hyperelastic. In this paper a constitutive model is derived to include the progressive reduction of the bulk stiffness and the permanent volumetric change of carbon-black filled elastomers subjected to quasi–static loading. The basis of the model is the theory of pseudo-elasticity, which including a softening variable modifies the dilatational strain energy function. An acceptable correspondence between the theory and the data is obtained.

Steady-state response of the parametrically excited axially moving string constituted by the Boltzmann superposition principle

Abstract: Summary. The steady-state transverse vibration of a parametrically excited axially moving string with geometric nonlinearity is investigated in this paper. The Boltzmann superposition principle is employed to characterize the material property of the string. The method of multiple scales is applied directly to the governing equation, which is a nonlinear partial-differential-integral equation. The solvability condition of eliminating the secular terms is established. Closed form solutions for the amplitude and the existence conditions of nontrivial steady-state response of the summation resonance are obtained. Some numerical examples showing effects of the viscoelastic parameter, the amplitude of excitation, the frequency of excitation, and the transport speed are presented.

The buckling of cross-ply laminated non-homogeneous orthotropic composite conical thin shells under a dynamic external pressure

Abstract: The subject of this investigation is to study the buckling of cross-ply laminated orthotropic truncated circular conical thin shells with variable Young's moduli and densities in the thickness direction, subjected to a uniform external pressure which is a power function of time. After obtaining the dynamic stability and compatibility equations we reduce both of them to a time dependent ordinary differential equation with variable coefficient by using Galerkin's method. The critical dynamic and static loading, the corresponding wave numbers, the dynamic factors, critical time and critical impulse are found analytically by applying the Ritz type variational method. The dynamic behavior of cross-ply laminated truncated conical shells is investigated with: (a) lamina that present variations in the Young's moduli and densities, (b) different numbers and ordering of layers, (c) variable semi-vertex angles, and (d) external pressures which vary with different powers of time. It is concluded that all these factors contribute to appreciable effects on the critical parameters of the problem in question.

On saturation-strip model of a permeable crack in a piezoelectric ceramic

Abstract: The saturation-strip model for piezoelectric crack is re-examined in a permeable environment to analyze fracture toughness of a piezoelectric ceramic. In this study, a permeable crack is modeled as a vanishing thin but finite rectangular slit with surface charge deposited along crack surfaces. This permeable saturation crack model reveals that there exists a possible leaky mode for electrical field, which allows applied electric field passing through the dielectric medium inside a crack. By taking into account the leaky mode effect, a first-order approximated solution is obtained with respect to slit height, h0, in the analysis of electrical and mechanical fields in the vicinity of a permeable crack tip. The permeable saturation crack model presented here also considers the effect of charge distribution on crack surfaces, which may be caused by any possible charge-discharge process in the dielectric medium inside the crack. A closed form solution is obtained for the permeable crack perpendicular to the poling direction under both mechanical as well electrical loads. Both local and global energy release rates are calculated. Remarkably, the global energy release rate for a permeable crack has an expression, where M is elastic modulus, a is the half crack length, e is permittivity constant, and e is piezoelectric constant. This result is in a broad agreement with some experimental observations and may be served as the fracture criterion for piezoelectric materials. This contribution elucidates how an applied electric field affects crack growth in piezoelectric ceramic through its interaction with permeable environment surrounding a crack.

Atomistic counterpart of micromorphic theory

Abstract: This paper aims to connect atomistic model to continuum theories. The fundamentals of micromorphic continuum theory are introduced. The instantaneous mechanical variables in an atomic system and the averaged quantities in a continuum field are derived and related to atomic variables. The balance laws of micromorphic continuum theory are formulated and verified from the viewpoint of molecular motion. Some conclusions and discussions are presented.

Modeling of uniaxial and biaxial ratcheting behavior of 1026 Carbon steel using the simplified Viscoplasticity Theory Based on Overstress (VBO)

Abstract: Hassan and Kyriakides [1], Hassan et al. [2], and Corona et al. [3] performed uniaxial and biaxial ratcheting experiments on heat-treated 1026 Carbon steel. The loading histories performed with uniaxial and tubular specimens were selected to simulate those encountered in nuclear reactor vessels. The stress-strain diagram of 1026 Carbon steel was used to determine the material constants in a simplified version of VBO. Small rate dependence was allowed. The model represents some nonlinear rate dependence, kinematic hardening and cyclically neutral behavior. The set of material constants determined only from uniaxial tests was used throughout the paper. Numerical experiments included are: (i) uniaxial stress-controlled cycling with various mean stresses, (ii) strain-controlled axial tests with tubular specimens under constant and variable internal pressure, and (iii) examination and variation of certain material constants of the VBO model that can influence ratcheting. Very good agreement with the experiments is found for the uniaxial case. However the ratchet strain accumulation during biaxial cycling is over-predicted by VBO and other constitutive models.

Nonlinear dynamic behavior of a piezothermoelastic laminated plate with anisotropic material properties

Abstract: In this paper, we analyze the nonlinear dynamic behavior of a piezothermoelastic laminated plate with anisotropic material properties. The analytical model is a rectangular laminate composed of fiber-reinforced laminae and piezoelectric layers. The model is assumed to be a symmetric cross-ply laminate with all egdes simply-supported and to be subjected to mechanical, thermal and electrical loads as intended control procedures or as disturbances. The von Karman strains are introduced to treat non-linear deformation. The behavior of the laminate is analyzed by using the Galerkin Method. We discuss the following quantities: (i) the buckling temperature due to in-plane thermal load; (ii) the large static deflection due to combined in-plane and anti-plane loads; (iii) the natural frequency of infinitesimal oscillation around the static equilibrium state; (iv) the natural frequency of the oscillation with finite amplitude around the static equilibrium state. Moreover, numerical examples are shown to investigate the methods to rise the buckling temperature, to linearize the thermal deflection and the natural frequencies by applying the electrical voltage to the piezoelectric actuators.

Quasi-static stress fields for a crack inclined to the property gradation in functionally graded materials

Abstract: Quasi-static stress fields for a crack inclined to the direction of property gradation in functionally graded materials (FGMs) are obtained through an asymptotic analysis coupled with Westergaard's stress function approach. The elastic modulus of the FGM is assumed to vary exponentially along the gradation direction. The mode mixity due to the inclination of the property gradient is accommodated in the analysis through superposition of opening and shear modes. The first four terms in the expansion of the stress field are derived to explicitly bring out the influence of nonhomogeneity on the structure of the stress field. Using these stress field contours of constant maximum shear stress, constant maximum principal stress, constant first stress invariant and constant out of plane displacement are generated, and the effect of inclination of the property gradation direction on these contours is discussed.

Propulsive performance and vortex shedding of a foil in flapping flight

Abstract: The propulsive performance and vortex shedding of an oscillating foil, which mimics biological locomotion, are investigated based on a computational fluid dynamics analysis. The objectives of this study are to investigate unsteady forces, in particular a thrust force, for the foil in pitching and plunging motion, and to deal with the relations of the propulsive performance with leading-edge vortex structure and vortex shedding in the near wake. The two-dimensional incompressible Navier–Stokes equations in the vorticity and stream-function formulation are solved by fourth-order essentially compact finite difference schemes for the space derivatives and a fourth-order Runge-Kutta scheme for the time advancement. To reveal the mechanism of the propulsive performance, the unsteady forces and the shedding of the leading- and trailing- edge vortices of the foil in the pitching and plunging motion are analyzed. Based on our calculated results, three types of the leading-edge vortex shedding, which have an effective influence on the vortex structures in the wake of the oscillating foil, are identified. The effects of some typical factors, such as the frequency and amplitude of the oscillation, the phase difference between the pitching and plunging motions, and the thickness ratio of the foil, on the vortex shedding and the unsteady forces are discussed.

Oscillatory Marangoni convection in a conducting fluid layer with a deformable free surface in the presence of a vertical magnetic field

Abstract: Linear stability theory is applied to the problem of the onset of oscillatory Marangoni convection in a horizontal layer of electrically conducting fluid heated from below in the presence of a vertical magnetic field. The fluid layer is bounded from below by a rigid boundary and from above by a deformable free surface. The critical Marangoni number M c, the critical wavenumber a c and the critical frequency ωc are obtained for wide ranges of the Prandtl number #E5/E5#1, the magnetic Prandtl number #E5/E5#2, the crispation number C r and the Chandrasekhar number Q. We present numerically a necessary and sufficient condition for oscillatory Marangoni convection to occur.

Effective bulk moduli of two functionally graded composites

Abstract: This paper considers the effective bulk moduli and the underlying elastic fields of a particle- and a fiber-reinforced composite whose matrix properties are graded linearly along the radial distance. The governing Fuchsian equations are first transformed into Riemann equations which, by means of change of the dependent variable, are further transformed into hypergeometric equations which are solved in terms of the hypergeometric function. This set of solutions provides an explicit dependence of the effective bulk behavior of the composites with arbitrary variations of the slope of the matrix moduli.

Bifurcations of parametric oscillations of beams with three equilibria

Abstract: Nonlinear curvature and nonlinear inertia are taken into account in the beam model. The constant part of the parametric force is proposed to be greater than the buckling one, therefore the beam has three equilibria. One-mode approximation of the beam oscillations is used. Bifurcations of the beam oscillations are analyzed by Melnikov's method. Moreover, beam oscillations close to the stable equilibriums are studied by the multiple scales method.