Noise influenced elastic cantilever dynamics with nonlinear tip interaction forces

In this work, the authors study the influence of noise on the dynamics of base-excited elastic cantilever structures at the macroscale and microscale by using experimental, numerical, and analytical means. The macroscale system is a base excited cantilever structure whose tip experiences nonlinear interaction forces. These interaction forces are constructed to be similar in form to tip interaction forces in tapping mode atomic force microscopy (AFM). The macroscale system is used to study nonlinear phenomena and apply the associated findings to the chosen AFM application. In the macroscale experiments, the tip of the cantilever structure experiences long-range attractive and short-range repulsive forces. There is a small magnet attached to the tip, and this magnet is attracted by another one mounted to a high-resolution translatory stage. The magnet fixed to the stage is covered by a compliant material that is periodically impacted by the cantilever’s tip. Building on their earlier work, wherein the authors showed that period-doubling bifurcations associated with near-grazing impacts occur during off-resonance base excitations of macroscale and microscale cantilevers, in the present work, the authors focus on studying the influence of Gaussian white noise when it is included as an addition to a deterministic base excitation input. The repulsive forces are modeled as Derjaguin–Muller–Toporov (DMT) contact forces in both the macroscale and microscale systems, and the attractive forces are modeled as van der Waals attractive forces in the microscale system and magnetic attractive forces in the macroscale system. A reduced-order model, based on a single mode approximation is used to numerically study the response for a combined deterministic and random base excitation. It is experimentally and numerically found that the addition of white Gaussian noise to a harmonic base excitation facilitates contact between the tip and the sample, when there was previously no contact with only the harmonic input, and results in a response that is nominally close to a period-doubled orbit. The qualitative change observed with the addition of noise is associated with near-grazing impacts between the tip and the sample. The numerical and experimental results further motivate the formulation of a general analytical framework, in which the Fokker–Planck equation is derived for the cantilever-impactor system. After making a set of approximations, the moment evolution equations are derived from the Fokker–Planck equation and numerically solved. The resulting findings support the experimental results and demonstrate that noise can be added to the input to facilitate contact between the cantilever’s tip and the surface, when there was previously no contact with only a harmonic input. The effects of Gaussian white noise are numerically studied for a tapping mode AFM application, and it is shown that contact between the tip and the sample can be realized by adding noise of an appropriate level to a harmonic excitation.     

Fragmentation of liquid and liquid-plastic media under unsteady strains

Based on a qualitative analysis of results of experimental studies, the main mechanisms of fragmentation of polar liquids and liquid-plastic media under dynamic loading are determined. In the case of low-viscosity liquids, such mechanisms are thermodynamic instability of foam, hydrodynamic instability of initial disturbances of the free surface, and the action of capillary forces. In the case of a polar high-viscosity liquid, the main mechanisms are shear instability of the structure, responsible for stratification of the medium along the lines of local failure of structural viscosity, and the action of capillary forces. The main mechanisms acting in liquid-plastic structured media (gels) are “spalling” in the zone of tensile stresses if the time of their formation is smaller than the time needed for the gel to transform to the sol state, as well as thermodynamic instability of foam and the action of capillary forces after the medium transition to the sol state. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 62–68, July–August, 2007.     

Nonstationary hydrodynamic characteristics of a double grid of profiles

The flow about a double grid of solid profiles of arbitrary shape which vibrate in a stream of an ideal incompressible fluid is considered. Behind the grid profiles, the nonstationary vortex traces simulated by the lines of contact velocity discontinuity are taken into account. The problem is reduced to the solution of a system of two integral equations relative to the fluid velocity on the initial profiles of the double grid under the assumption that the vibration amplitudes are small. Formulas for calculating the nonstationary forces and moments are derived. The dependences of these forces on the shape, mutual positions, and laws of vibration of the grid profiles are studied. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 150–155, July–August, 1999.     

Effect of High-Frequency Vibration on Convection in Miscible Fluids

The effect of high-frequency vibrations of the field of external mass forces on convection in miscible fluids is considered for a system of convection equations obtained by an averaging technique. The structures of flows formed under initial conditions corresponding to physical experiments in microgravity are examined. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 2, pp. 49–59, March–April, 2006.     

Vortex diffraction on a swept wing

The forces acting on a swept wing in the presence of a vortex induced by a delta wing, as well as the velocity field in the vicinity of the swept wing, have been measured. By means of the “frozen,” vortex model and a specially-developed numerical panel method, the forces and moments acting on the wing are calculated from the known velocity field. Comparison of the calculated and measured force characteristics makes it possible to determine the extent to which the model fits the physical flow pattern. It is shown that for the intense vortex considered in this study the model gives results which disagree sharply with the experimental data. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 98–105, November–December, 1998. The study was supported by the International Scientific and Technical Center under grant No. 201.     

Comparisons of rotordynamic coefficients in stepped labyrinth seals by using Colebrook-White friction factor model

The objective of this work is to observe the effects of friction factors for the stepped labyrinth seals. The gas flow through the seals creates net pressure and shear forces acting on the rotor. It is necessary to predict these forces for reliably operating turbomachinery. So we investigated the effect of shear forces on the calculation of rotordynamic coefficients by comparing the results in the case shear forces are considered and in the case they are neglected. We also compared our results, obtained with the Colebrook–White friction factor model, with some reference experimental and computational results.     

Some primal problems of the axisymmetric theory of elasticity for space with a flat slit bounded by a circle

We consider the first and second primal problems of the axisymmetric theory of elasticity for space with a round slit and a mixed problem in which forces are specified on one side of the slit and displacements are specified on the other side. The problems reduce to conjugation problems for generalized analytic functions on rectilinear segments, whose solution is obtained in closed form. Institute of Applied Mechanics, Russian Academy of Sciences, Moscow 117334. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 1 pp. 144–151, January–February, 2000.     

Numerical analysis of the stability of nonisothermal couette flow

The stability of convective motion of a liquid between two rotating heated cylinders is investigated in the absence of external forces. The mathematical model for describing the convection is obtained from the general equations [1, 2] on the assumption that the density of the liquid, the thermal conductivity, the specific heat and the viscosity coefficients depend only on temperature, and that the work done by the pressure forces and the viscous dissipation are negligibly small. The thermal expansion coefficient of the liquid is not assumed to be small, which distinguishes the models in question from the classical Oberbeck-Boussinesq model [1, 3, 4]. Rostov-on-Don. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 70–76, September–October, 1988.     

The impact of non-DLVO forces on the onset of shear thickening of concentrated electrically stabilized suspensions

This paper exposes an extension of an activation model previously published by the authors. When particles arranged along the compression axis of a sheared suspension, they may overcome the electrostatic repulsion and form force chains associated with shear thickening. A percolation-based consideration allows an estimation of the impact of the force chains on a flowing suspension. It suggests that similar to mode coupling models, the suspension becomes unstable before the critical stress evaluated from the activation model is reached. The percolated force chains lead to discontinuous shear thickening. The model predictions are compared with results from two experimental studies on aqueous suspensions of inorganic oxides; in one of them, hydration repulsion and in the other hydrophobic attraction can be expected. It is shown that the incorporation of non-Derjaguin–Landau–Verwey–Overbeek forces greatly improve predictions of the shear thickening instability.     

Limiting velocity of crack propagation in elastic materials

A crack is represented as a continuous set of linear dislocations. Simple analytical expressions are obtained for the potential and kinetic energies of the environment of moving cracks and the attached mass of cracks for an arbitrary form of the stress applied to the crack P(x). It is shown that the indicated analytical expressions are bilinear integrals of the functions P(x) and ∂P(x)/∂x. These integrals are calculated in explicit form for a constant stress over the entire crack length and the stress due to the action of molecular adhesion forces in a narrow region near the crack openings. It is shown that the calculation results does not depend on the form of molecular adhesion forces. The proposed approach to describing cracks and calculations based on it has made it possible for the first time to obtain a complete analytical expression for the limiting crack propagation velocity in elastic materials as a function of the main mechanical characteristics of such materials. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 158–166, July–August, 2009.     

Azimuthal wave motions on the surface of a rotating fluid cylinder

Wave motions in a fluid cylinder rotating about the axis are investigated within the framework of the linear theory. The cylinder is assumed to be fairly long. This makes it possible to restrict attention to the study of the plane oscillation pattern. The fluid is assumed to be ideal and incompressible. The models in which the fluid particles are confined by gravitational (body) or/and capillary forces (surface stress forces) are considered. A mode analysis is carried out and the dispersion relations are constructed. Traveling and steady-state waves on the surface of the fluid cylinder are investigated; qualitative effects ("wave inertia") are established. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 128–133, May–June, 1998. The work was carried out with financial support from the Russian Foundation for Basic Research (project No. 96-01-00221). An erratum to this article is available at .     

An Internal Damping Model for the Absolute Nodal Coordinate Formulation

Introducing internal damping in multibody system simulations is important as real-life systems usually exhibit this type of energy dissipation mechanism. When using an inertial coordinate method such as the absolute nodal coordinate formulation, damping forces must be carefully formulated in order not to damp rigid body motion, as both this and deformation are described by the same set of absolute nodal coordinates. This paper presents an internal damping model based on linear viscoelasticity for the absolute nodal coordinate formulation. A practical procedure for estimating the parameters that govern the dissipation of energy is proposed. The absence of energy dissipation under rigid body motion is demonstrated both analytically and numerically. Geometric nonlinearity is accounted for as deformations and deformation rates are evaluated by using the Green–Lagrange strain–displacement relationship. In addition, the resulting damping forces are functions of some constant matrices that can be calculated in advance, thereby avoiding the integration over the element volume each time the damping force vector is evaluated.     

Hydrodynamic interaction of particles

The hydrodynamic interaction of solid particles in a viscous incompressible fluid is considered. An analytical solution of the problem of interaction of two or more particles is obtained. The forces and torques exerted on the particles and also the linear and angular particle velocities are calculated. A comparison with the results obtained earlier is given. Saransk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 112–119, March–April, 1998. The work was supported by a grant from the Norweigan Research Council.     

Theoretical identification of forces resisting longitudinal movement of drillstrings in curved wells

The equations of balance of external and internal forces acting on a drillstring in a curved well are set up based on the formulations of direct and inverse problems in the mechanics of flexible curvilinear rods. The lowering, lifting, and rotation of a drillstring are studied. A method is proposed to calculate the internal longitudinal force, the forces of interaction between the drillstring and the well wall, and the forces of friction. An example is considered. It is shown that even small geometrical imperfections of the well path have a significant effect on the balance of external and internal forces __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 11, pp. 80–89, November 2007.     

In vivo mechanical study of helical cardiac pacing electrode interacting with canine myocardium

Cardiac pacing is a medical device to help human to overcome arrhythmia and to recover the regular beats of heart. A helical configuration of electrode tip is a new type of cardiac pacing lead distal tip. The helical electrode attaches itself to the desired site of heart by screwing its helical tip into the myocardium. In vivo experiments on anesthetized dogs were carried out to measure the acute interactions between helical electrode and myocardium during screw-in and pull-out processes. These data would be helpful for electrode tip design and electrode/myocardium adherence safety evaluation. They also provide reliability data for clinical site choice of human heart to implant and to fix the pacing lead. A special design of the helical tip using strain gauges is instrumented for the measurement of the screw-in and pull-out forces. We obtained the data of screw-in torques and pull-out forces for five different types of helical electrodes at nine designed sites on ten canine hearts. The results indicate that the screw-in torques increased steplike while the torque–time curves presente saw-tooth fashion. The maximum torque has a range of 0.3–1.9 N mm. Obvious differences are observed for different types of helical tips and for different test sites. Large pull-out forces are frequently obtained at epicardium of left ventricle and right ventricle lateral wall, and the forces obtained at right ventricle apex and outflow tract of right ventricle are normally small. The differences in pull-out forces are dictated by the geometrical configuration of helix and regional structures of heart muscle.     

Mixed Convection in a Vertical Porous Channel

A numerical study of mixed convection in a vertical channel filled with a porous medium including the effect of inertial forces is studied by taking into account the effect of viscous and Darcy dissipations. The flow is modeled using the Brinkman–Forchheimer-extended Darcy equations. The two boundaries are considered as isothermal–isothermal, isoflux–isothermal and isothermal–isoflux for the left and right walls of the channel and kept either at equal or at different temperatures. The governing equations are solved numerically by finite difference method with Southwell–Over–Relaxation technique for extended Darcy model and analytically using perturbation series method for Darcian model. The velocity and temperature fields are obtained for various porous parameter, inertia effect, product of Brinkman number and Grashof number and the ratio of Grashof number and Reynolds number for equal and different wall temperatures. Nusselt number at the walls is also determined for three types of thermal boundary conditions. The viscous dissipation enhances the flow reversal in the case of downward flow while it counters the flow in the case of upward flow. The Darcy and inertial drag terms suppress the flow. It is found that analytical and numerical solutions agree very well for the Darcian model. An erratum to this article is available at .     

Time-dependent interaction between two inviscid fluid flows separated by a semi-infinite non-rigid plate in a flat-walled channel

The problem of the time-dependent interaction between two inviscid weightless fluids separated by a semi-infinite non-rigid plate in a channel with fixed rigid walls is solved in the linear approximation. The general case of deformation and harmonic oscillations of the plate (flapping mover) are considered. The time-dependent hydrodynamic reaction forces, the position of the interface, and the dynamic characteristics of the mover are determined. Kazan’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 67–76, January–February, 1997.     

Flow of electrolytes in a porous medium

A two-scale model of ion transfer in a porous medium is obtained for one-dimensional horizontal flows under the action of a pressure gradient and an external electric field by the method of homogenization. Steady equations of electroosmotic flows in flat horizontal nano-sized slits separated by thin dielectric partitions are averaged over a small-scale variable. The resultant macroequations include Poisson’s equation for the vertical component of the electric field and Onsager’s relations between flows and forces. The total horizontal flow rate of the fluid is found to depend linearly on the pressure gradient and external electric field, and the coefficients in this linear relation are calculated with the use of microequations. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 4, pp. 162–173, July–August, 2008.     

Track dynamic behavior at rail welds at high speed

As a vehicle passing through a track with different weld irregularities, the dynamic performance of track com- ponents is investigated in detail by using a coupled vehi- cle-track model. In the model, the vehicle is modeled as a multi-body system with 35 degrees of freedom, and a Timoshenko beam is used to model the rails which are dis- cretely supported by sleepers. In the track model, the sleepers are modeled as rigid bodies accounting for their vertical, lat- eral and rolling motions and assumed to move backward at a constant speed to simulate the vehicle running along the track at the same speed. In the study of the coupled vehicle and track dynamics, the Hertizian contact theory and the theory proposed by Shen-Hedrick-Elkins are, respectively, used to calculate normal and creep forces between the wheel and the rails. In the calculation of the normal forces, the coefficient of the normal contact stiffness is determined by transient contact condition of the wheel and rail surface. In the calcu- lation of the creepages, the lateral, roll-over motions of the rail and the fact that the relative velocity between the wheel and rail in their common normal direction is equal to zero are simultaneously taken into account. The motion equations of the vehicle and track are solved by means of an explicit integration method, in which the rail weld irregularities are modeled as local track vertical deviations described by some ideal cosine functions. The effects of the train speed, the axle load, the wavelength and depth of the irregularities, and the weld center position in a sleeper span on the wheel-rail impact loading are analyzed. The numerical results obtained are greatly useful in the tolerance design of welded rail pro- file irregularity caused by hand-grinding after rail welding and track maintenances.     

Dynamic contact angle for etting of a surface by a van der waals fluid

We consider the creeping motion of a thin layer of a nonvolatile viscous fluid spreading due to capillary forces over a rigid surface covered by a thin homogeneous film (microfilm). The influence of van der Waals forces on the asymptotic slope of the free boundary of the layer is studied in the region of large thickness, where capillary forces dominate. A solution of the problem of the slope angle is obtained for the entire possible range of the microfilm thickness. In the limit of small thickness of the microfilm, this solution is in agreement with the well-known solution of the problem of the dynamics of wetting of a dry surface in the presence of a precursory film and van der Waals forces. The role of the condition at the end of the precursory film is studied. Institute of Mechanics of Multiphase Systems, Siberian Division, Russian Academy of Sciences. Tyumen' 625000. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika. Vol. 41, No. 4, pp. 101–105, July–August. 2000.