A note on the contact problem in an ultrasonic travelling wave motor

In this paper we discuss the non-linear contact problem between the stator and the rotor of an ultrasonic travelling were motor. For a first simplified mathematical model the problem is formulated for a linear motor in which the stator is modelled as a Bernoulli-Euler beam and the slider (rotor) is assumed to be rigid. A thin layer of visco-elastic material is assumed to exist between stator and slider. Expressions are obtained for the contact pressure between the two parts. The frictional forces both in the sticking and in the sliding zone can then be easily obtained assuming Coulomb friction.     

Two-dimensional elastica analysis of equilibrium shapes of single-anchor inflatable dams

Several types of inflatable dams are considered. These are long, air-inflated, cylindrical structures on a rigid foundation. Sometimes one of the long edges of a sheet is folded back to the other edge, and then the two edges are clamped to the foundation along a single anchoring line. A second configuration can be modeled as two sheets attached along two long edges, with one edge anchored and the other free to lift as air pressure is applied between the sheets. Another device treated here is a hinged spillway gate lifted by an inflatable bladder. The cross section of the dam or bladder is analyzed as an inextensible elastica. The governing equations and boundary conditions are formulated for each case, and shooting methods are utilized to obtain numerical solutions for the equilibrium shapes. The effects of the internal air pressure and the external water height are investigated.     

A general solution of the axisymmetric contact problem for biphasic cartilage layers

A unilateral axisymmetric contact problem for articular cartilage layers is considered. The articular cartilages bonded to subchondral bones are modeled as biphasic materials consisting of a solid phase and a fluid phase. It is assumed that the subchondral bones are rigid and shaped like bodies of revolution with arbitrary convex profiles. The obtained closed-form analytical solution is valid over time periods compared with the typical diffusion time and can be used for increasing loading.     

Finite inflation of a hyperelastic toroidal membrane over a cylindrical rim

The present paper is devoted to the study of finite inflation of a hyperelastic toroidal membrane on a cylindrical rim under uniform internal pressure. Both compliant and rigid frictionless rims have been considered. The compliant cylindrical rim is modeled as a linear distributed stiffness. The initial cross-section of the torus is assumed to be circular, and the membrane material is assumed to be a homogeneous and isotropic Mooney–Rivlin solid. The problem is formulated as a two point boundary value problem and solved using a shooting method by employing the Nelder–Meads search technique. The optimization function is constructed on a two (three) dimensional search space for the compliant cylinder (rigid cylinder). The effect of the inflation pressure, material properties and elastic properties of the rim on the state of stretch and stress, and on the geometry of the inflated torus have been studied, and some interesting results have been obtained. The stability of the inflated configurations in terms of occurrence of the impending wrinkling state in the membrane has also been studied.     

A damage constitutive model for sliding friction coupled to wear

Several new constitutive models are formulated for the planar interface of a soft body sliding on a rigid soil, describing stick-slip phenomena due to friction, and wear due to abrasion. Attention is focused on damage at the interface, by neglecting any interaction with damage of body and any propagation of damage inside the body. Models are formulated in the general framework of the Thermodynamics of the irreversible processes and account for suitable defined internal variables of phenomenological type, namely gap, isotropic friction hardening and wear. The main feature of the new presented models is that the formulation of the wear process at the interface is obtained in the contest of Damage Mechanics, and it is based on the formal analogy between abrasion of a soft body and ductile damage of an elastic-plastic material. By following this approach, a scalar wear field, an effective stress and appropriate state and dissipation potentials are defined, able to describe a tangential isotropic wear process due to stick-slip and to hardening mechanism. Both cases of linear and nonlinear friction hardening are formulated; moreover, wearable and no-wearable bodies are considered. Numerical results relevant to one-dimensional problems are illustrated for monotonic, forward-backward and cyclic displacement time-histories, showing evolution for stress, gap and wear. Results furnished by different models are compared and discussed.     

From Non-Linear Elasticity to Linearized Theory: Examples Defying Intuition

Material frame indifference implies that the solution in non-linear elasticity theory for a connected body rigidly rotated at its border is a rigid, stress-free, deformation. If the same problem is considered within linear elasticity theory, considered as an approximation to the true elastic situation, one should expect that if the angle of rotation is small, the body still undergoes a rigid deformation while the corresponding stress, though not zero, remains consistently small. Here, we show that this is true, in general, only for homogeneous bodies. Counterexamples of inhomogeneous bodies are presented for which, whatever small the angle of rotation is, the linear elastic solution is by no means a rigid rotation (in a particular case it is an “explosion”) while the stress may even become infinite. If the same examples are re-interpreted as problems in an elasticity theory based upon genuinely linear constitutive relations which retain their validity also for finite deformations, it is shown that they would deliver constraint reaction forces that are not in equilibrium in the actual, deformed, state. This furnishes another characterization of the impossibility of an exact linear constitutive theory for elastic solids with zero residual stress.      

On the pulsating flow behavior of a biological fluid: human blood

In this work, the rectilinear flow of a complex fluid (human blood) under a pulsating time-dependent pressure gradient is analyzed. A first approximation of the real case of blood flowing in a vein is described. The normalized pressure gradient simulates the pumping work of the heart while the flow geometry (circular tube) is assumed rigid, smooth, and cylindrical. The rheological behavior of blood with different cholesterol levels is modeled using the Bautista–Manero–Puig (BMP) constitutive equation. According to the analytical solution, a flow enhancement is predicted to first order which represents the optimum pumping work of the heart which governs the flow of blood in the entire body. This work is a contribution to the understanding of the complex rheology involved in the discontinuous pressure-driven flow of blood in the human body.     

Efficiency of a vibroprotection system with an isochronous roller damper

Low-frequency vibrations of a vibroprotection “roller damper-movable bearing body” system of rigid bodies under the action of an external harmonic excitation are considered. The working surface of the damper working body is formed by a brachistochrone. The dynamic equations of common no-slip motion of the damper working body on a hinged roller and of the bearing body are formulated. The roller damper tuning parameters are determined.     

The Finite Element Analysis of Interaction Problem Through Water,Soil, Balloon and Pile

This paper presents the finite element analysis of an interaction problem involving water, soil, balloon and pile. A building with friction piles is considered, and balloons are introduced to the top of piles. To control the vertical displacement of the building, water is injected into or removed from the balloons. The two-dimensional incompressible Navier-Stokes equation is introduced, and the ALE (Arbitrary Lagrangian Eulerian) method is applied to the water flow analysis. The FS (Fractional Step) method is also applied in the finite element formulation. The soil, which is assumed as a linear elastic body, is subjected to the deformation analysis. The balloon and pile are assumed as a linear elastic truss and a rigid frame, and the deformation analysis is also performed. All the components are discretized by the finite element method in space and are interactively solved by taking into account continuity conditions of traction and displacement.     

Large Motions of a Moored Floating Breakwater Modeled as an Impact Oscillator

Two-dimensional motions of a floating breakwater moored to thesea floor by two cables are considered. The breakwater is modeled bothas a point mass and as a rigid body. The mooring lines are assumed tohave no effect on the breakwater when they are slack, and to provide aninstantaneous impulsive force when they become taut, analogous to animpact oscillator or a ball bouncing on a rigid surface. The axialcomponent of the velocity is reduced at this instantaneous tautcondition. Fluid inertia and damping are not included, and the waveforces are assumed to be harmonic. A critical force is defined, and theeffects of the forcing frequency, the coefficient of restitution, andthe shape and size of the body on the critical force are examined.Trajectories of the motion are plotted and the impact velocities arecomputed and analyzed. Knowledge of the number and magnitude of theseimpacts is useful in assessing fatigue of the mooring lines.     

Application of the nonsmooth dynamics approach to model and analysis of the contact-impact events in cam-follower systems

The dynamic modeling and analysis of planar rigid multibody systems that experience contact-impact events is presented and discussed throughout this work. The methodology is based on the nonsmooth dynamics approach, in which the interaction of the colliding bodies is modeled with multiple frictional unilateral constraints. Rigid multibody systems are stated as an equality of measures, which are formulated at the velocity-impulse level. The equations of motion are complemented with constitutive laws for the forces and impulses in the normal and tangential directions. In this work, the unilateral constraints are described by a set-valued force law of the type of Signorini??s condition, while the frictional contacts are characterized by a set-valued force law of the type of Coulomb??s law for dry friction. The resulting contact-impact problem is formulated and solved as an augmented Lagrangian approach, which is embedded in the Moreau time-stepping method. The effectiveness of the methodologies presented in this work is demonstrated throughout the dynamic simulation of a cam-follower system of an industrial cutting file machine.     

Structural analogies on systems of deformable bodies coupled with non-linear layers

The paper is addressed at phenomenological mapping and mathematical analogies of oscillatory regimes in systems of coupled deformable bodies. Systems consist of coupled deformable bodies like plates, beams, belts or membranes that are connected through visco-elastic non-linear layer, modeled by continuously distributed elements of Kelvin–Voigt type with nonlinearity of third order. Using the mathematical analogies the similarities of structural models in systems of plates, beams, belts or membranes are obvious. The structural models consist by a set of two coupled non-homogenous partial non-linear differential equations. The problems to solve are divided into space and time domains by the classical Bernoulli–Fourier method. In the time domains the systems of coupled ordinary non-linear differential equations are completely analog for different systems of deformable bodies and are solved by using the Krilov–Bogolyubov–Mitropolskiy asymptotic method. This paper presents the beauty of mathematical analytical calculus which could be the same even for physically different systems.The mathematical numerical calculus is a powerful and useful tool for making the final conclusions between many input and output values. The conclusions about nonlinear phenomena in multi-body systems dynamics have been revealed from the particular example of double plate׳s system stationary and non-stationary oscillatory regimes.     

Mechanical experiments to identify homogeneous bodies

All bodies are inhomogeneous at some scale but experience has shown that some of these bodies can be idealized as a homogeneous body. Here we examine which bodies can be idealized as a homogeneous body when they are subjected to a non-dissipative mechanical process. This is done by studying circumstances in which an inhomogeneous body admits pure stretch homogeneous deformations. Then, we devise experiments wherein these circumstances are prevented. If homogeneous deformation is observed in these devised experiments, the body could be modeled as a homogeneous body. We limit our analysis to a class of isotropic elastic bodies deforming from a stress free reference configuration whose Cauchy stress is explicitly related to left Cauchy–Green deformation tensor. It is further assumed that the constitutive relation is differentiable function of the position vector of material particles in the stress free reference configuration. Then, we find that a cuboid made of compressible and isotropic material could be modeled as a homogeneous body if it deforms homogeneously due to the application of the normal stresses on all of its six faces and the magnitude of the normal stresses on three orthogonal faces are different. A cuboid made of incompressible and isotropic material could be modeled as a homogeneous body, if it deforms homogeneously in two different biaxial experiments, such that the plane in which the forces are applied in the two biaxial experiments is mutually orthogonal.     

A homogenized viscoelastic model for masonry structures

A linear viscous model for evaluating the stresses and strains produced in masonry structures over time is presented. The model is based on rigorous homogenization procedures and the following two assumptions: that the structure is composed of either rigid or elastic blocks, and that the mortar is viscoelastic. The hypothesis of rigid block is particularly suitable for historical masonry, in which stone blocks may be assumed as rigid bodies, while the hypothesis of elastic blocks may be assumed for newly constructed brickwork structures. The hypothesis of viscoelastic mortar is based on the observation that non-linear phenomena may be concentrated in mortar joints. Under these assumptions, constitutive homogenized viscous functions are obtained in an analytical form.Some meaningful cases are discussed: masonry columns subject to minor and major eccentricity, and a masonry panel subject to both horizontal and vertical loads. The major eccentricity case is analysed taking into account both the effect of viscosity and the no-tension hypothesis, whereas the bi-dimensional loading case is analysed to verify the sensitivity of masonry behaviour to viscous function. In the masonry wall considered, the principal stresses are both of compression, and the no-tension assumption may therefore be discounted.     

Effect of the order of the finite element and selection of the constrained modes in deformable body dynamics

Finite elements with different orders can be used in the analysis of constrained deformable bodies that undergo large rigid body displacements. The constrained mode shapes resulting from the use of finite elements with different orders differ in the way the stiffness of the body bending and extension are defined. The constrained modes also depend on the selection of the boundary conditions. Using the same type of finite element, different sets of boundary conditions lead to different sets of constrained modes. In this investigation, the effect of the order of the element as well as the selection of the constrained mode shapes is examined numerically. To this end, the constant strain three node triangular element and the quadratic six node triangular element are used. The results obtained using the three node triangular element are compared with the higher order six node triangular element. The equations of motion for the three and six node triangular elements are formulated from assumed linear and quadratic displacement fields, respectively. Both assumed displacement fields can describe large rigid body translational and rotational displacements. Consequently, the dynamic formulation presented in this investigation can also be used in the large deformation analysis. Using the finite element displacement field, the mass, stiffness, and inertia invariants of the three and six-node triangular elements are formulated. Standard finite element assembly techniques are used to formulate the differential equations of motion for mechanical systems consisting of interconnected deformable bodies. Using a multibody four bar mechanism, numerical results of the different elements and their respective performance are presented. These results indicate that the three node triangular element does not perform well in bending modes of deformation.     

Strain and fracture of circular plates under static and dynamical loading by a spherical body

The strain and fracture of plates under the action of a load normal to their planes was studied in numerous papers. A review of publications in this field in the case of impact by a freely flying body is given in [1–3]. At first, researchers’ attention was mainly paid to the so-called ideal version of collision in which the normal impact of a rigid body on the plate center was considered and the boundary conditions did not affect the results of impact. The plate strains were studied near and in the region of impact, the minimal velocities were determined for a body of some specific shape for which the plate is punched through (the so-called ballistic limit); the shapes of fractured punched plates and the residual velocity of the body if its initial velocity exceeds the ballistic limit were also determined. In the last years, the more complicated cases of collision have been studied, namely, the case in which the impact is not directed along the normal to the plate plane and the impact velocity vector does not coincide with the body symmetry axis as well as the case of impacts on shells. The research in this field was represented in [2, 3]. But in this case the influence of the boundary conditions is still considered insufficiently. This gap was indicated in [2].In the present paper, we study the normal impacts of spherical bodies and deformable cylindrical bodies with spherical heads on circular plates for various boundary conditions and mechanical characteristics of their material. We consider the plate strains, determine the impact velocity at which the plate is punched through, and clarify the mechanism and the sequence of the plate fracture and break-though depending on their mechanical characteristics and boundary conditions. We make an attempt to perform numerical studies of the dynamic deflection at the center of a plate fixed on the boundary using its experimentally determined quasistatic rigidity and taking into account the boundary conditions for determining the associated mass. We estimate the influence of the body mass on the ballistic limit. The use of rigid spherical bodies permits treating any variations in the results of impacts as a characteristic reaction of the plates themselves, because in this case it is unnecessary to deal with the body orientation with respect to the velocity vector. For impacts with such bodies, we used plates made of aluminum alloys and of lead. We studied how the strength of cylindrical bodies with spherical heads made of plasticine or lead affects the strain of plates made of AMTsM alloy.     

Finite element analysis of grain-by-grain deformation by crystal plasticity with couple stress

Rigid–plastic crystal plasticity with the rate-sensitive constitutive behavior of a slip system has been formulated within the framework of a two-dimensional finite element method to predict the grain-by-grain deformation of single- and polycrystalline FCC metals. For that purpose, individual grains are represented by several numbers of finite elements to describe the sub-grain deformation behavior, and couple stress has been introduced into the equilibrium equation to be able to describe the size effect as well as to prevent mesh-dependent predictions. A modified virtual work-rate principle with an approximate interface constraint has been suggested to use a C ⁰-continuous element in the finite element implementation, and the couple stress work-rate has been formulated on the basis of an assumed constitutive behavior. Simulated plane-strain compressions of a single crystal cube show that the shearing and the deformation load are closely related to the imbedded lattice orientation of the crystal grain, and that the sub-grain deformation and the load magnitude can be controlled by the couple stress hardening. It is also confirmed that almost the same predictions are obtained for different mesh systems by considering the couple stress hardening. Simulated plane-strain compressions of a bi-crystal show considerably curved grain-by-grain surface profiles after large reduction for several combinations of the imbedded lattice orientation. The high couple stress hardening predicted around grain boundaries is supposed to be related to the grain size effect. It is also supposed that consideration of couple stress is necessary to predict the sub-grain or the grain-by-grain deformation, and the couple stress hardening may be used to describe the state of microstructures in grain.     

Simulation of peristaltic flow of chyme in small intestine for couple stress fluid

In this article couple stress fluid have been considered for the peristaltic flow of chyme in intestine. Problem under consideration have been formulated assuming that two non-periodic sinusoidal wave of different wavelength propagate with same speed c along the outer wall of the tube. Governing equations have been simplified under the assumptions of long wavelength and low Reynolds number approximation (such assumption are consistent that Re (Reynold number) is very small and long wavelength approximation also exists in the small intestine). Exact solutions have been evaluated for velocity and pressure rise. Physical behaviour of different parameter of couple stress fluid have been presented graphically for velocity, pressure rise, pressure gradient and frictional forces. The stream lines are also made against different parameters.     

DYNAMIC MODELING FOR AIRSHIP EQUIPPED WITH BALLONETS AND BALLAST

IntroductionAirships have an enormous, yet untapped potential as low-speed, or even steadyplatforms for aerial exploration, monitoring, and surveillance, as well for transportation andtelecommunication purposes[1]. Similar to the underwater vehicle, airsh…     

Criterion of multiple collisions in a simple mechanical system with viscous damping and dry friction

A criterion has been formulated for single and multiple central collisions of two rigid bodies of a simple mechanical system that can be either conservative or non-conservative. Analytical discussion has been confined to investigations of the possibility of appearance of four successive collisions between the bodies with damping neglected and two collisions when viscous damping and Coulomb dry friction are considered.