Showing papers in "International Journal of Non-linear Mechanics in 2005"

Stokes’ first problem for a second grade fluid in a porous half-space with heated boundary

Abstract: Based on a modified Darcy's law, Stokes’ first problem was investigated for a second grade fluid in a porous half-space with a heated flat plate. Exact solutions of the velocity and temperature fields were obtained using Fourier sine transforms. In contrast to the classical Stokes’ first problem, there is a steady-state solution for the second grade fluid in the porous half-space, which is a damping exponential function with respect to the distance from the flat plate. The well-known solutions for Newtonian fluids in non-porous or porous half-space appear in limiting cases of our solutions.

Peridynamic modeling of membranes and fibers

Abstract: The peridynamic theory of continuum mechanics allows damage, fracture, and long-range forces to be treated as natural components of the deformation of a material. In this paper, the peridynamic approach is applied to small thickness two- and one-dimensional structures. For membranes, a constitutive model is described appropriate for rubbery sheets that can form cracks. This model is used to perform numerical simulations of the stretching and dynamic tearing of membranes. A similar approach is applied to one-dimensional string like structures that undergrow stretching, bending, and failure. Long-range forces similar to van der Waals interactions at the nanoscale influence the equilibrium configurations of these structures, how they deform, and possibly self-assembly.

An unsteady analysis of non-Newtonian blood flow through tapered arteries with a stenosis

Abstract: The problem of non-Newtonian and nonlinear blood flow through a stenosed artery is solved numerically where the non-Newtonian rheology of the flowing blood is characterised by the generalised Power-law model. An improved shape of the time-variant stenosis present in the tapered arterial lumen is given mathematically in order to update resemblance to the in vivo situation. The vascular wall deformability is taken to be elastic (moving wall), however a comparison has been made with nonlinear visco-elastic wall motion. Finite difference scheme has been used to solve the unsteady nonlinear Navier–Stokes equations in cylindrical coordinates system governing flow assuming axial symmetry under laminar flow condition so that the problem effectively becomes two-dimensional. The present analytical treatment bears the potential to calculate the rate of flow, the resistive impedance and the wall shear stress with minor significance of computational complexity by exploiting the appropriate physically realistic prescribed conditions. The model is also employed to study the effects of the taper angle, wall deformation, severity of the stenosis within its fixed length, steeper stenosis of the same severity, nonlinearity and non-Newtonian rheology of the flowing blood on the flow field. An extensive quantitative analysis is performed through numerical computations of the desired quantities having physiological relevance through their graphical representations so as to validate the applicability of the present model.

Mechanical response of fiber-reinforced incompressible non-linearly elastic solids

Abstract: The mechanical response of some fiber-reinforced incompressible non-linearly elastic solids is examined under homogeneous deformation. In particular, the materials under consideration are neo-Hookean models augmented with a function that accounts for the existence of a unidirectional reinforcement. This function endows the material with its anisotropic character and is referred to as a reinforcing model. The nature of the anisotropy considered has a particular influence on the shear response of the material, in contrast to previous analyses in which the reinforcing model was taken to depend only on the stretch in the fiber direction.

Effects of radiation and heat source on MHD flow of a viscoelastic liquid and heat transfer over a stretching sheet

Abstract: We study the MHD flow and also heat transfer in a viscoelastic liquid over a stretching sheet in the presence of radiation. The stretching of the sheet is assumed to be proportional to the distance from the slit. Two different temperature conditions are studied, namely (i) the sheet with prescribed surface temperature (PST) and (ii) the sheet with prescribed wall heat flux (PHF). The basic boundary layer equations for momentum and heat transfer, which are non-linear partial differential equations, are converted into non-linear ordinary differential equations by means of similarity transformation. The resulting non-linear momentum differential equation is solved exactly. The energy equation in the presence of viscous dissipation (or frictional heating), internal heat generation or absorption, and radiation is a differential equation with variable coefficients, which is transformed to a confluent hypergeometric differential equation using a new variable and using the Rosseland approximation for the radiation. The governing differential equations are solved analytically and the effects of various parameters on velocity profiles, skin friction coefficient, temperature profile and wall heat transfer are presented graphically. The results have possible technological applications in liquid-based systems involving stretchable materials.

Experimental study of non-linear energy pumping occurring at a single fast frequency

Abstract: Experimental verification of passive non-linear energy pumping in a two-degree-of-freedom system comprising a damped linear oscillator coupled to an essentially non-linear attachment is carried out. In the experiments presented the non-linear attachment interacts with a single linear mode and, hence, energy pumping occurs at a single ‘fast’ frequency in the neighborhood of the eigenfrequency of the linear mode. Good agreement between simulated and experimental results was observed, in spite of the strongly (essentially) non-linear and transient nature of the dynamics of the system considered. The experiments bear out earlier predictions that a significant fraction of the energy introduced directly to a linear structure by an external impulsive (broadband) load can be transferred (pumped) to an essentially non-linear attachment, and dissipated there locally without spreading back to the system. In addition, the reported experimental results confirm that (a) non-linear energy pumping in systems of coupled oscillators can occur only above a certain threshold of the input energy, and (b) there is an optimal value of the energy input at which a maximum portion of the energy is absorbed and dissipated at the NES.

Elastic instabilities in rubber

Abstract: Materials that undergo large elastic deformations can exhibit novel instabilities. Several examples are considered here: development of an aneurysm on inflating a cylindrical rubber tube; non-uniform stretching on inflating a spherical balloon; expansion of small cavities in rubber blocks when they are subjected to a critical amount of triaxial tension or when they are supersaturated with a dissolved gas; wrinkling of the surface of a block at a critical amount of compression; and the sudden formation of “knots” on twisting stretched cylindrical rods. These various deformations are analyzed in terms of simple strain energy functions using Rivlin's theory of large elastic deformations. The theoretical results are then compared with experimental measurements of the onset of unstable states. Such comparisons provide new tests of Rivlin's theory and, at least in principle, critical tests of proposed strain energy functions for rubber. Moreover, the onset of highly non-uniform deformations has serious implications for the fatigue life and fracture resistance of rubber components.

Peristaltic transport of a Herschel–Bulkley fluid in an inclined tube

Abstract: Peristaltic flow of Herschel–Bulkley fluid in an inclined tube is analyzed. The velocity distribution, the stream function and the volume flow rate are obtained. Also, when the yield stress ratio τ → 0 , and when the inclination parameter α = 0 and the fluid parameter n = 1 , the results agree with those of Jaffrin and Shapiro (Ann. Rev. Fluid Mech. 3 (1971) 13) for peristaltic transport of a Newtonian fluid in a horizontal tube. The effects of τ and n on the pressure drop and the mean flow are discussed through graphs. Furthermore, the results for the peristaltic transport of Bingham and power law fluids through a flexible tube are obtained and discussed. The results obtained for the flow characteristics reveal many interesting behaviors that warrant further study of the effects of Herschel–Bulkley fluid on the flow characteristics.

Sakiadis flow of an upper-convected Maxwell fluid

Abstract: The flow of an upper-convected Maxwell (UCM) fluid is studied theoretically above a rigid plate moving steadily in an otherwise quiescent fluid. It is assumed that the Reynolds number of the flow is high enough for boundary layer approximation to be valid. Assuming a laminar, two-dimensional flow above the plate, the concept of stream function coupled with the concept of similarity solution is utilized to reduce the governing equations into a single third-order ODE. It is concluded that the fluid's elasticity destroys similarity between velocity profiles; thus an attempt was made to find local similarity solutions. Three different methods will be used to solve the governing equation: (i) the perturbation method, (ii) the fourth-order Runge–Kutta method, and (iii) the finite-difference method. The velocity profiles obtained using the latter two methods are shown to be virtually the same at corresponding Deborah number. The velocity profiles obtained using perturbation method, in addition to being different from those of the other two methods, are dubious in that they imply some degree of reverse flow. The wall skin friction coefficient is predicted to decrease with an increase in the Deborah number for Sakiadis flow of a UCM fluid. This prediction is in direct contradiction with that reported in the literature for a second-grade fluid.

Twist buckling and the foldable cylinder: an exercise in origami

Abstract: Basic mechanisms for the buckling of a thin cylindrical shell under torsional loading are reviewed from a post-buckling perspective. Deflections are considered so far into the large-deflection range that the shell is allowed to fold to a flat two-dimensional form, in a mechanism reminiscent of a deployable structure. Critical and initial post-buckling effects are explored through concepts of energy minimization and hidden symmetries. For comparisons with the final large-deflection folded shape, a truss element program is employed. It is shown that, as buckling develops, the mode shape must change to accommodate both the symmetry-breaking aspects of the predominately inwards deflection, and the rotation of peak and valley lines of the buckle pattern necessary to accommodate the geometry of the final folded shape.

Non-linear vibrations of doubly curved shallow shells

Abstract: Large amplitude (geometrically non-linear) vibrations of doubly curved shallow shells with rectangular base, simply supported at the four edges and subjected to harmonic excitation normal to the surface in the spectral neighbourhood of the fundamental mode are investigated. Two different non-linear strain–displacement relationships, from the Donnell's and Novozhilov's shell theories, are used to calculate the elastic strain energy. In-plane inertia and geometric imperfections are taken into account. The solution is obtained by Lagrangian approach. The non-linear equations of motion are studied by using (i) a code based on arclength continuation method that allows bifurcation analysis and (ii) direct time integration. Numerical results are compared to those available in the literature and convergence of the solution is shown. Interaction of modes having integer ratio among their natural frequencies, giving rise to internal resonances, is discussed. Shell stability under static and dynamic load is also investigated by using continuation method, bifurcation diagram from direct time integration and calculation of the Lyapunov exponents and Lyapunov dimension. Interesting phenomena such as (i) snap-through instability, (ii) subharmonic response, (iii) period doubling bifurcations and (iv) chaotic behaviour have been observed.

One-to-two global-local interaction in a cable-stayed beam observed through analytical, finite element and experimental models

Abstract: The mechanism of cable end angle-variation induced oscillations in the non-linear interactions between beams and cables in stayed-systems is first explained by a proposed analytical model. It is then verified by both experimental and finite element models. The non-linear interaction maximizes its effects for cable oscillations when inherent quadratic coupling between local and global modes produces energy transfer from low to high frequency vibrations by means of a one-to-two global–local autoparametric resonance. The response of the analytical model is fully described using a continuation method applied directly to the reduced two degree of freedom discrete model showing that, for a selected one-to-two global–local resonant system, primary harmonic excitation of the global mode produces large oscillations of the local mode at twice the excitation frequency. Detailed comparisons between the responses of the analytical model, experimental results and finite element simulations show excellent agreement both in the qualitative behaviour and in the calculated/measured response amplitudes.

Non-linear interactions in imperfect beams at veering

Abstract: Non-linear interactions in a hinged–hinged uniform moderately curved beam with a torsional spring at one end are investigated. The two-mode interaction is a one-to-one autoparametric resonance activated in the vicinity of veering of the frequencies of the lowest two modes and results from the non-linear stretching of the beam centerline. The excitation is a base acceleration that is involved in a primary resonance with either the first mode only or with both modes. The ensuing non-linear responses and their stability are studied by computing force– and frequency–response curves via bifurcation analysis tools. Both the sensitivity of the internal resonance detuning—the gap between the veering frequencies—and the linear modal structure are investigated by varying the rise of the beam half-sinusoidal rest configuration and the torsional spring constant. The internal and external resonance detunings are varied accordingly to construct the non-linear system response curves. The beam mixed-mode response is shown to undergo several bifurcations, including Hopf and homoclinic bifurcations, along with the phenomenon of frequency island generation and mode localization.

Non-linear dynamic response of a rotating radial Timoshenko beam with periodic pulse loading at the free-end

Abstract: Consideration is given to the dynamic response of a Timoshenko beam under repeated pulse loading. Starting with the basic dynamical equations for a rotating radial cantilever Timoshenko beam clamped at the hub in a centrifugal force field, a system of equations are derived for coupled axial and lateral motions which includes the transverse shear and rotary inertia effects, as well. The hyperbolic wave equation governing the axial motion is coupled with the flexural wave equation governing the lateral motion of the beam through the velocity-dependent skew-symmetric Coriolis force terms. In the analytical formulation, Rayleigh–Ritz method with a set of sinusoidal displacement shape functions is used to determine stiffness, mass and gyroscopic matrices of the system. The tip of the rotating beam is subjected to a periodic pulse load due to local rubbing against the outer case introducing Coulomb friction in the system. Transient response of the beam with the tip deforming due to rub is discussed in terms of the frequency shift and non-linear dynamic response of the rotating beam. Numerical results are presented for this vibro-impact problem of hard rub with varying coefficients of friction and the contact-load time. The effects of beam tip rub forces transmitted through the system are considered to analyze the conditions for dynamic stability of a rotating blade with intermittent rub.

Some results for generalized neo-Hookean elastic materials

Abstract: Several deformations of non-linear elastic materials are used to study the implications of the strain energy density function being dependent only on the first strain invariant. Two kinds of results are obtained, those that compare responses with and without dependence on the second invariant, and those specific to materials whose strain energy functions depend only on the first strain invariant. The deformations are (i) homogenous biaxial extension, (ii) shear superposed on triaxial extension, (iii) inflation of a circular membrane, (iv) circular shear superimposed on a press fit cylinder, (v) torsion of a circular cylinder.

The construction of non-linear normal modes for systems with internal resonance

Abstract: A numerical method, based on the invariant manifold approach, is presented for constructing non-linear normal modes for systems with internal resonances. In order to parameterize the non-linear normal modes of interest, multiple pairs of system state variables involved in the internal resonance are kept as ‘seeds’ for the construction of the multi-mode invariant manifold. All the remaining degrees of freedom are then constrained to these ‘seed’, or master, variables, resulting in a system of non-linear partial differential equations that govern the constraint relationships, and these are solved numerically. The computationally-intensive solution procedure uses a combination of finite difference schemes and Galerkin-based expansion approaches. It is illustrated using two examples, both of which focus on the construction of two-mode models. The first example is based on the analysis of a simple three-degree-of-freedom example system, and is used to demonstrate the approach. An invariant manifold that captures two non-linear normal modes is constructed, resulting in a reduced order model that accurately captures the system dynamics. The methodology is then applied to a larger order system, specifically, an 18-degree-of-freedom rotating beam model that features a three-to-one internal resonance between the first two flapping modes. The accuracy of the non-linear two-mode reduced order model is verified by comparing time-domain simulations of the two DOF model and the full system equations of motion.

Flow and heat transfer of an electrically conducting fluid of second grade in a porous medium over a stretching sheet subject to a transverse magnetic field

Abstract: An analysis is performed for flow and heat transfer of a steady laminar boundary layer flow of an electrically conducting fluid of second grade in a porous medium subject to a transverse uniform magnetic field past a semi-infinite stretching sheet with power-law surface temperature or power-law surface heat flux. The effects of viscous dissipation, internal heat generation of absorption and work done due to deformation are considered in the energy equation. The variations of surface temperature gradient for the prescribed surface temperature case (PST) and surface temperature for the prescribed heat flux case (PHF) with various parameters are tabulated. The asymptotic expansions of the solutions for large Prandtl number are also given for the two heating conditions. It is shown that, when the Eckert number is large enough, the heat flow may transfer from the fluid to the wall rather than from the wall to the fluid when Eckert number is small. A physical explanation is given for this phenomenon.

Unsteady flows of oldroyd-b fluids in a channel of rectangular cross section

Abstract: The aim of this note is to present the exact solutions corresponding to two types of unsteady flows of an Oldroyd-B fluid in a channel of rectangular cross-section. The solutions that have been obtained satisfy both the associate partial differential equations and all imposed initial and boundary conditions. For λ r or λ → 0 they tend toward similar solutions for a Maxwell or second-grade fluid. If both λ r and λ → 0 , the solutions for Navier–Stokes fluids are recovered.

Periodic sticking motion in a two-degree-of-freedom impact oscillator

Abstract: Periodic sticking motions can occur in vibro-impact systems for certain parameter ranges. When the coefficient of restitution is low (or zero), the range of periodic sticking motions can become large. In this work the dynamics of periodic sticking orbits with both zero and non-zero coefficient of restitution are considered. The dynamics of the periodic orbit is simulated as the forcing frequency of the system is varied. In particular, the loci of Poincare fixed points in the sticking plane are computed as the forcing frequency of the system is varied. For zero coefficient of restitution, the size of the sticking region for a particular choice of parameters appears to be maximized. We consider this idea by computing the sticking region for zero and non-zero coefficient of restitution values. It has been shown that periodic sticking orbits can bifurcate via the rising/multi-sliding bifurcation. In the final part of this paper, we describe three types of post-bifurcation behavior which occur for the zero coefficient of restitution case. This includes two types of rising bifurcation and a border orbit crossing event.

Detection and description of non-linear phenomena in experimental modal analysis via linearity plots

Abstract: Ground vibration tests (GVTs) on aircraft prototypes are mainly performed to experimentally identify the structural dynamic behaviour in terms of a modal model This assumes a linear dynamic behaviour of the structure However, in the practice of ground vibration testing it is often observed that structures do not behave in a perfectly linear manner Non-linearities can be determined, for example, by free play in junctions, hydraulic systems in control surfaces, or friction This paper compiles measured, typical, non-linear phenomena from various GVTs on large aircraft The standard procedure in GVTs nowadays is the application of the Harmonic Balance method which linearizes the dynamic behaviour on the level of excitation The procedure requires a harmonic excitation of the structure which is usually performed during phase resonance testing The non-linear behaviour is investigated in terms of linearity plots in which the resonance frequency of a mode is plotted as a function of the excitation level The experimental data is then compatible with all post-processing procedures for the measured results, eg updating of the finite element model or flutter calculations This paper shows measured linearity plots for some typical non-linear phenomena In the second part of the paper analytical linearity plots for different non-linear stiffness and damping models are considered in order to investigate whether the type of non-linearity can be identified from measured linearity plots The analytical linearity plots are discussed with respect to their application limits The analytical linearity plots are used to interpret the experimental linearity plots stemming from various GVTs on different aircraft prototypes Finally, the observability of non-linear stiffness and non-linear damping characteristics via linearity plots is assessed

Hopf–Hopf bifurcation and invariant torus of a vibro-impact system

Abstract: Hopf–Hopf bifurcation of a three-degree-of-freedom vibro-impact system is considered in this paper. The period n - 1 motion is determined and its Poincare map is established. When two pairs of complex conjugate eigenvalues of the Jacobian matrix of the map at fixed point cross the unit circle simultaneously, the six-dimensional Poincare map is reduced to its four-dimensional normal form by the center manifold and the normal form methods. Two-parameter unfoldings and bifurcation diagrams near the critical point are analyzed. It is proved that there exist the torus T 1 and T 2 bifurcation under some parameter combinations. Numerical simulation results reveal that the vibro-impact system may present different types of complicated invariant tori T 1 and T 2 as two controlling parameters varying near Hopf–Hopf bifurcation points. Investigating torus bifurcation in vibro-impact system has important significance for studying global dynamical behavior and routes to chaos via quasi-period bifurcation.

Buckling behavior of compression-loaded composite cylindrical shells with reinforced cutouts

Abstract: Results from a numerical study of the response of thin-walled compression-loaded quasi-isotropic laminated composite cylindrical shells with unreinforced and reinforced square cutouts are presented. The effects of cutout reinforcement orthotropy, size, and thickness on the nonlinear response of the shells are described. A nonlinear analysis procedure has been used to predict the nonlinear response of the shells. The results indicate that a local buckling response occurs in the shell near the cutout when subjected to load and is caused by a nonlinear coupling between local shell-wall deformations and in-plane destabilizing compression stresses near the cutout. In general, reinforcement around a cutout in a compression-loaded shell is shown to retard or eliminate the local buckling response near the cutout and increase the buckling load of the shell. However, some results show that certain reinforcement configurations can cause an unexpected increase in the magnitude of local deformations and stresses in the shell and cause a reduction in the buckling load. Specific cases are presented that suggest that the orthotropy, thickness, and size of a cutout reinforcement in a shell can be tailored to achieve improved buckling response characteristics.

Magnetohydrodynamic mixed convection in a vertical channel

Abstract: The problem of combined free and forced convective magnetohydrodynamic flow in a vertical channel is analysed by taking into account the effect of viscous and ohmic dissipations. The channel walls are maintained at equal or at different constant temperatures. The velocity field and the temperature field are obtained analytically by perturbation series method and numerically by finite difference technique. The results are presented for various values of the Brinkman number and the ratio of Grashof number to the Reynolds number for both equal and different wall temperatures. Nusselt number at the walls is determined. It is found that the viscous dissipation enhances the flow reversal in the case of downward flow while it counters the flow in the case of upward flow. It is also found that the analytical and numerical solutions agree very well for small values of e .

Elastic–plastic contact model for rough surfaces based on plastic asperity concept

Abstract: A mathematical formulation for the contact of rough surfaces is presented. The derivation of the contact model is facilitated through the definition of plastic asperities that are assumed to be embedded at a critical depth within the actual surface asperities. The surface asperities are assumed to deform elastically whereas the plastic asperities experience only plastic deformation. The deformation of plastic asperities is made to obey the law of conservation of volume. It is believed that the proposed model is advantageous since (a) it provides a more accurate account of elastic–plastic behavior of surfaces in contact and (b) it is applicable to model formulations that involve asperity shoulder-to-shoulder contact. Comparison of numerical results for estimating true contact area and contact force using the proposed model and the earlier methods suggest that the proposed approach provides a more realistic prediction of elastic–plastic contact behavior.

Uniformity of stresses inside an elliptic inclusion in finite plane elastostatics

Abstract: We consider an elastic inclusion embedded in a particular class of harmonic materials subjected to uniform remote stress. Using complex variable techniques, we show that if the Piola stress within the inclusion is uniform, the inclusion is necessarily an ellipse except in the special case when the (uniform) remote stress assumes a particular form. In addition, we obtain the complete solution for an elliptic inclusion with uniform interior stress for any uniform remote stress distribution.

Non-linear response of laminated composite plates under thermomechanical loading

Abstract: The non-linear response of laminated composite plates under thermomechanical loading is studied using the third-order shear deformation theory (TSDT) that includes classical and first-order shear deformation theories (CLPT and FSDT) as special cases. Geometric non-linearity in the von Karman sense is considered. The temperature field is assumed to be uniform in the plate. Layers of magnetostrictive material, Terfenol-D, are used to actively control the center deflection. The negative velocity feedback control is used with the constant gain value. The effects of lamination scheme, magnitude of loading, layer material properties, and boundary conditions are studied under thermomechanical loading.

Non-linear strain rate dependent micro-mechanical composite material model for finite element impact and crashworthiness simulation

Abstract: The present study aims at implementation of a strain rate dependent, non-linear, micro-mechanics material model for laminated, unidirectional polymer matrix composites into the explicit finite element code LSDYNA. The objective is to develop an accurate and simple micro-mechanical, rate dependent material model, which is computationally efficient. Within the model a representative volume cell is assumed. The stress–strain relation including rate dependent effects for the micro-model is derived for both shell elements and solid elements. Micro-failure criterion is presented for each material constituent and failure mode. The implemented model can deal with problems such as impact, crashworthiness, and failure analysis under quasi-static loads. The developed material model has a wide range of applications such as jet engine jackets, armor plates, and structural crashworthiness simulation. The deformation response of two representative composite materials with varying fiber orientation is presented using the described technique. The predicted results compare favorably to experimental values.

On unsteady unidirectional flows of a second grade fluid

Abstract: Some properties of unsteady unidirectional flows of a fluid of second grade are considered for flows produced by the sudden application of a constant pressure gradient or by the impulsive motion of one or two boundaries. Exact analytical solutions for these flows are obtained and the results are compared with those of a Newtonian fluid. It is found that the stress at the initial time on the stationary boundary for flows generated by the impulsive motion of a boundary is infinite for a Newtonian fluid and is finite for a second grade fluid. Furthermore, it is shown that initially the stress on the stationary boundary, for flows started from rest by sudden application of a constant pressure gradient is zero for a Newtonian fluid and is not zero for a fluid of second grade. The required time to attain the asymptotic value of a second grade fluid is longer than that for a Newtonian fluid. It should be mentioned that the expressions for the flow properties, such as velocity, obtained by the Laplace transform method are exactly the same as the ones obtained for the Couette and Poiseuille flows and those which are constructed by the Fourier method. The solution of the governing equation for flows such as the flow over a plane wall and the Couette flow is in a series form which is slowly convergent for small values of time. To overcome the difficulty in the calculation of the value of the velocity for small values of time, a practical method is given. The other property of unsteady flows of a second grade fluid is that the no-slip boundary condition is sufficient for unsteady flows, but it is not sufficient for steady flows so that an additional condition is needed. In order to discuss the properties of unsteady unidirectional flows of a second grade fluid, some illustrative examples are given.

Non-linear shear vibrations of piezoceramic actuators

Abstract: A wide range of non-linear effects are observed in piezoceramic materials. For small stresses and weak electric fields, piezoceramics are usually described by linearized constitutive equations around an operating point. However, typical non-linear vibration behavior is observed at weak electric fields near resonance frequency excitations of the piezoceramics. This non-linear behavior is observed in terms of a softening behavior and the decrease of normalized amplitude response with increase in excitation voltage. In this paper the authors have attempted to model this behavior using higher order cubic conservative and non-conservative terms in the constitutive equations. Two-dimensional kinematic relations are assumed, which satisfy the considered reduced set of constitutive relations. Hamilton's principle for the piezoelectric material is applied to obtain the non-linear equation of motion of the piezoceramic rectangular parallelepiped specimen, and the Ritz method is used to discretize it. The resulting equation of motion is solved using a perturbation technique. Linear and non-linear parameters for the model are identified. The results from the theoretical model and the experiments are compared. The non-linear effects described in this paper may have strong influence on the design of the devices, e.g. ultrasonic motors, which utilize the piezoceramics near the resonance frequency excitation.

Brachistochrone with Coulomb friction

Abstract: The classical brachistochrone is considered with the inclusion of a resistant force, which is due to Coulomb friction, in addition to the uniform gravitational force that is present. The solution to this problem is expressed in terms of standard functions, and it is developed in two separate ways by means of constrained variational calculus methods. These ways involve formulations of the problem in terms of temporal and spatial independent variables, respectively. The equations of motion that result in both cases are non-linear and coupled. The utilization of path variables is a central feature of the developments provided.