Showing papers in "International Journal of Non-linear Mechanics in 2007"

Peristaltic flow and heat transfer in a vertical porous annulus, with long wave approximation

Abstract: In this paper, we study the interaction of peristalsis with heat transfer for the flow of a viscous fluid in a vertical porous annular region between two concentric tubes. Long wavelength approximation (that is, the wavelength of the peristaltic wave is large in comparison with the radius of the tube) is used to linearise the governing equations. Using the perturbation method, the solutions are obtained for the velocity and the temperature fields. Also, the closed form expressions are derived for the pressure–flow relationship and the heat transfer at the wall. The effect of pressure drop on flux is observed to be almost negligible for peristaltic waves of large amplitude; however, the mean flux is found to increase by 10–12% as the free convection parameter increases from 1 to 2. Also, the heat transfer at the wall is affected significantly by the amplitude of the peristaltic wave. This warrants further study on the effects of peristalsis on the flow and heat transfer characteristics.

A new branch of solutions of boundary-layer flows over a permeable stretching plate

Abstract: The steady-state boundary-layer flows over a permeable stretching sheet are investigated by an analytic method for strongly non-linear problems, namely the homotopy analysis method (HAM). Two branches of solutions are obtained. One of them agrees well with the known numerical solutions. The other is new and has never been reported in general cases. The entrainment velocity of the new branch of solutions is always smaller than that of the known ones. For permeable stretching sheet with sufficiently large suction of mass flux, the difference between the shear stresses and velocity profiles of two branches of solutions is obvious: the shear stress of the new branch of solutions is considerably larger than that of the known ones. However, for impermeable sheet and permeable sheet with injection or small suction of mass flux, the shear stress and the velocity profile of two branches of solutions are rather close: in some cases the difference is so small that the new branch of solutions might be neglected even by numerical techniques. This reveals the reason why the new branch of solutions has not been reported. This work also illustrates that, for some non-linear problems having multiple solutions, analytic techniques are sometimes more effective than numerical methods.

Non-linear dynamics of a cracked cantilever beam under harmonic excitation

Abstract: The presence of cracks in a structure is usually detected by adopting a linear approach through the monitoring of changes in its dynamic response features, such as natural frequencies and mode shapes. But these linear vibration procedures do not always come up to practical results because of their inherently low sensitivity to defects. Since a crack introduces non-linearities in the system, their use in damage detection merits to be investigated. With this aim the present paper is devoted to analysing the peculiar features of the non-linear response of a cracked beam. The problem of a cantilever beam with an asymmetric edge crack subjected to a harmonic forcing at the tip is considered as a plane problem and is solved by using two-dimensional finite elements; the behaviour of the breathing crack is simulated as a frictionless contact problem. The modification of the response with respect to the linear one is outlined: in particular, excitation of sub- and super-harmonics, period doubling, and quasi-impulsive behaviour at crack interfaces are the main achievements. These response characteristics, strictly due to the presence of a crack, can be used in non-linear techniques of crack identification.

Lyapunov exponents as a criterion for the dynamic pull-in instability of electrostatically actuated microstructures

Abstract: The dynamic pull-in instability of double clamped microscale beams actuated by a suddenly applied distributed electrostatic force and subjected to non-linear squeeze film damping is investigated. A reduced order model is built using the Galerkin decomposition with undamped linear modes as base functions and verified through comparison with numerical finite differences solution. The stability analysis of a beam actuated by one and two electrodes symmetrically located at two sides of the beam and operated by a step-input voltage is performed by evaluating the largest Lyapunov exponent, the sign of which defines the character of the response. It is shown that this approach provides an efficient quantitative criterion for the evaluation of dynamic pull-in instability, especially when combined with compact reduced order models. Based on the Lyapunov exponent criterion, the influence of various parameters on the beam dynamic stability is investigated.

Low-frequency band gaps in chains with attached non-linear oscillators

Abstract: The aim of this article is to investigate the wave propagation in one-dimensional chains with attached non-linear local oscillators by using analytical and numerical models. The focus is on the influence of non-linearities on the filtering properties of the chain in the low frequency range. Periodic systems with alternating properties exhibit interesting dynamic characteristics that enable them to act as filters. Waves can propagate along them within specific bands of frequencies called pass bands, and attenuate within bands of frequencies called stop bands or band gaps. Stop bands in structures with periodic or random inclusions are located mainly in the high frequency range, as the wavelength has to be comparable with the distance between the alternating parts. Band gaps may also exist in structures with locally attached oscillators. In the linear case the gap is located around the resonant frequency of the oscillators, and thus a stop band can be created in the lower frequency range. In the case with non-linear oscillators the results show that the position of the band gap can be shifted, and the shift depends on the amplitude and the degree of non-linear behaviour.

Non-linear vibrations and frequency response analysis of piezoelectrically driven microcantilevers

Abstract: Microcantilevers have recently received widespread attentions due to their extreme applicability and versatility in both biological and non-biological applications. Along this line, this paper undertakes the non-linear vibrations of a piezoelectrically driven microcantilever beam as a common configuration in many scanning probe microscopy and nanomechanical cantilever biosensor systems. A part of the microcantilever beam surface is covered by a piezoelectric layer (typically ZnO), which acts both as an actuator and sensor. The bending vibrations of the microcantilever beam are studied considering the inextensibility condition and the coupling between electrical and mechanical properties in the piezoelectric materials. The non-linear terms appear in the form of quadratic expression due to presence of piezoelectric layer, and cubic form due to geometrical non-linearities. The Galerkin approximation is then utilized to discretize the equations of motion. In addition, the method of multiple scales is applied to arrive at the closed form solution for the fundamental natural frequency of the system. An experimental setup consisting of a commercial piezoelectric microcantilever attached on the stand of a state-of-the-art microsystem analyzer for non-contact vibration measurement is utilized to verify the theoretical developments. It is found that the experimental results and theoretical findings are in good agreement, which demonstrates that the non-linear modeling framework could provide a better dynamic representation of the microcantilever than the previous linear models. Due to microscale nature of the system, excitation amplitude plays an important role since even a small change in the amplitude of excitation can lead to significant vibrations and frequency shift.

Finite deformations of fibre-reinforced elastic solids with fibre bending stiffness

Abstract: In the conventional theory of finite deformations of fibre-reinforced elastic solids it is assumed that the strain-energy is an isotropic invariant function of the deformation and a unit vector A that defines the fibre direction and is convected with the material. This leads to a constitutive equation that involves no natural length. To incorporate fibre bending stiffness into a continuum theory, we make the more general assumption that the strain-energy depends on deformation, fibre direction, and the gradients of the fibre direction in the deformed configuration. The resulting extended theory requires, in general, a non-symmetric stress and the couple-stress. The constitutive equations for stress and couple-stress are formulated in a general way, and specialized to the case in which dependence on the fibre direction gradients is restricted to dependence on their directional derivatives in the fibre direction. This is further specialized to the case of plane strain, and finite pure bending of a thick plate is solved as an example. We also formulate and develop the linearized theory in which the stress and couple-stress are linear functions of the first and second spacial derivatives of the displacement. In this case for the symmetric part of the stress we recover the standard equations of transversely isotropic linear elasticity, with five elastic moduli, and find that, in the most general case, a further seven moduli are required to characterize the couple-stress.

Zener internal damping in modelling of axially moving viscoelastic beam with time-dependent tension

Abstract: Non-linear vibrations of axially moving beam with time-dependent tension are investigated in this paper. The beam material is modelled as three-parameter Zener element. The Galerkin method and the fourth order Runge–Kutta method are used to solve the governing non-linear partial-differential equation. The effects of the transport speed, the tension perturbation amplitude and the internal damping on the dynamic behaviour of the system are numerically investigated. The Poincare maps and bifurcation diagrams are constructed to classify the vibrations. For small values of the transport speed and the amplitude of periodic perturbation the system is asymptotically stable with its response tending to zero. With the increase of parameters one can observe the coexistence of attractors. Regular and chaotic motion occur when the internal damping increases.

The response of clamped–clamped microbeams under mechanical shock

Abstract: We present modeling, simulation, and characterization for the dynamic response of clamped–clamped microbeams under mechanical shock. A Galerkin-based reduced-order model is utilized and its results are verified by comparing to finite-element results. The results indicate that the response of a microbeam to mechanical shock is inherently non-linear because of the dominating effect of mid-plane stretching. The effect of the shock pulse shape is investigated. It is concluded that the shape of the shock pulse can result in significant dynamic amplification in the response of the microbeam even in cases where the shock load is considered quasi-static. The combined effect of the electrostatic force and mechanical shock is investigated. The results show that this combined effect can lead to early instability in microelectromechanical systems (MEMS) devices through dynamic pull-in. This could explain some of the reported experimental evidences for the existence of strange modes of failure of MEMS devices under mechanical shock and impact. These failures are characterized by overlaps between moving microstructures and stationary electrodes, which cause electrical shorts. The shock-electrostatic interaction is shown to be promising to design smart MEMS switches triggered at predetermined level of shock and acceleration. Finally, the mechanical shock combined with the packaging effect of MEMS devices is analyzed. A single-degree-of-freedom model representing the motion of the package, which is mounted over a printed circuit board, coupled with the continuous beam model is utilized. Our results reveal that neglecting the effect of the package motion on the response of microbeams can overestimate or underestimate their response. It is concluded that a poor design of the package may result in severe amplification of the shock effect leading to a device failure.

An incompressible SPH method for simulation of unsteady viscoelastic free-surface flows

Abstract: In this paper, an incompressible smoothed particle hydrodynamics (SPH) method is presented to solve unsteady free-surface flows. Both Newtonian and viscoelastic fluids are considered. In the case of viscoelastic fluids, both the Maxwell and Oldroyd-B models are investigated. The proposed SPH method uses a Poisson pressure equation to satisfy the incompressibility constraints. The solution algorithm is an explicit predictor-corrector scheme and employs an adaptive smoothing length based on density variations. To alleviate the numerical difficulties encountered when fluid is highly stretched, an artificial stress term is incorporated into the momentum equation which reduces the risk of unrealistic fractures in the material. Two challenging test cases, the impacting drop and the jet buckling problems, are solved to demonstrate the capability of the proposed scheme in handling viscoelastic flows with complex free surfaces. The jet buckling test case was solved for a wide range of Weissenberg numbers. It was shown that in all cases the method is stable and fairly accurate and agrees well with the available data.

An analytical study of linear and non-linear double diffusive convection with Soret and Dufour effects in couple stress fluid

Abstract: The onset of double diffusive convection in a two component couple stress fluid layer with Soret and Dufour effects has been studied using both linear and non-linear stability analysis. The linear theory depends on normal mode technique and non-linear analysis depends on a minimal representation of double Fourier series. The effect of couple stress parameter, the Soret and Dufour parameters, and the Prandtl number on the stationary and oscillatory convection are presented graphically. The Dufour parameter enhances the stability of the couple stress fluid system in case of both stationary and oscillatory mode. The effect of positive Soret parameter is to destabilize the system in case of stationary mode while it stabilizes the system in case of oscillatory mode. The negative Soret parameter enhances the stability in both stationary and oscillatory mode. The couple stress parameter enhances the stability of the system in both stationary and oscillatory modes. The Dufour parameter increases the heat transfer while the couple stress parameter has reverse effect. The Soret parameter has negligible influence on heat transfer. Both Dufour and Soret parameters increases the mass transfer while the couple stress parameter has dual effect depending on the value of the Rayleigh number.

Synchronization and phase relations in the motion of two-pendulum system

Abstract: The synchronization phenomenon in non-linear oscillating system is studied by means of examination of the coupled pendulums. Dependence of the phase shift between pendulum states on system parameters and initial conditions is studied both analytically and numerically. The harmonic linearization technique is applied for analytical examinations.

Theory of amplitude modulation atomic force microscopy with and without Q-Control

Abstract: The present text reviews the fundamentals of amplitude-modulation atomic force microscopy (AM-AFM), which is frequently also referred to as dynamic force microscopy, non-contact atomic force microscopy, or “tapping mode” AFM. It is intended to address two different kinds of readerships. First, due to a thorough coverage of the theory necessary to explain the basic features observed in AM-AFM, it serves theoreticians that would like to gain overview on how nanoscale cantilevers interacting with the surrounding environment can be used to characterize nanoscale features and properties of suitable sample surfaces. On the other hand, it is designed to introduce experimentalists to the physics underlying AM-AFM measurements to a degree that is not too specialized, but sufficient to allow them measuring the quantities they need with optimized imaging parameters. More specifically, this article first covers the basics of the various driving mechanisms that are used in AFM imaging modes relying on oscillating cantilevers. From this starting point, an analytical theory of AM-AFM is developed, which also includes the effects of external resonance enhancement (“Q-Control”). This theory is then applied in conjunction with numerical simulations to various situations occurring while imaging in air or liquids. In particular, benefits and drawbacks of driving exactly at resonance frequency are examined as opposed to detuned driving. Finally, a new method for the continuous measurement of the tip–sample interaction force is discussed.

Effects of a crack on the stability of a non-linear rotor system

Abstract: The stability of a rotor system presenting a transverse breathing crack is studied by considering the effects of crack depth, crack location and the shaft's rotational speed. The harmonic balance method, in combination with a path-following continuation procedure, is used to calculate the periodic response of a non-linear model of a cracked rotor system. The stability of the rotor's periodic movements is studied in the frequency domain by introducing the effects of a perturbation on the periodic solution for the cracked rotor system. It is shown that the areas of instability increase considerably when the crack deepens, and that the crack's position and depth are the main factors affecting not only the non-linear behaviour of the rotor system but also the different zones of dynamic instability in the periodic solution for the cracked rotor. The effects of some other system parameters (including the disk position and the stiffness of the supports) on the dynamic stability of the non-linear periodic response of the cracked rotor system are also investigated.

On the mathematical theory of vehicular traffic flow II: Discrete velocity kinetic models

Abstract: This paper deals with the modelling of vehicular traffic flow by methods of the discrete mathematical kinetic theory. The discretization is developed in the velocity space by a grid adapted to the local density. The discretization overcomes, at least in part, some technical difficulties related to the selection of the correct representation scale, while the adaptative grid allows an improved description of various phenomena related to vehicular traffic flow. Specific models are proposed and a qualitative and computational analysis is developed to show the properties of the model and their ability to describe real flow conditions. A critical analysis, proposed in the last part of the paper, outlines suitable research perspectives.

A principle of virtual work for combined electrostatic and mechanical loading of materials

Abstract: The equations governing mechanics and electrostatics are formulated for a system in which the material deformations and electrostatic polarizations are arbitrary. A mechanical/electrostatic energy balance is formulated for this situation in terms of the electric enthalpy, in which the electric potential and the electric field are the independent variables, and charge and electric displacement, respectively, are the conjugate thermodynamic forces. This energy statement is presented in the form of a principle of virtual work (PVW), in which external virtual work is equated to internal virtual work. The resulting expression involves an internal material virtual work in which (1) material polarization is work-conjugate to increments of electric field, and (2) a combination of Cauchy stress, Maxwell stress and a product of polarization and electric field is work-conjugate to increments of strain. This PVW is valid for all material types, including those that are conservative and those that are dissipative. Such a virtual work expression is the basis for a rigorous formulation of a finite element method for problems involving the deformation and electrostatic charging of materials, including electroactive polymers and switchable ferroelectrics. The internal virtual work expression is used to develop the structure of conservative constitutive laws governing, for example, electroactive elastomers and piezoelectric materials, thereby determining the form of the Maxwell or electrostatic stress. It is shown that the Maxwell or electrostatic stress has a form fully constrained by the constitutive law and cannot be chosen independently of it. The structure of constitutive laws for dissipative materials, such as viscoelastic electroactive polymers and switchable ferroelectrics, is similarly determined, and it is shown that the Maxwell or electrostatic stress for these materials is identical to that for a material having the same conservative response when the dissipative processes in the material are shut off. The form of the internal virtual work is used further to develop the structure of dissipative constitutive laws controlled by rearrangement of material internal variables.

Elastic instabilities for strain-stiffening rubber-like spherical and cylindrical thin shells under inflation

Abstract: This paper is concerned with investigation of the effects of strain-stiffening on the classical limit point instability that is well-known to occur in the inflation of internally pressurized rubber-like spherical thin shells (balloons) and circular cylindrical thin tubes composed of incompressible isotropic non-linearly elastic materials. For a variety of specific strain-energy densities that give rise to strain-stiffening in the stress-stretch response, the inflation pressure versus stretch relations are given explicitly and the non-monotonic character of the inflation curves is examined. While such results are known for constitutive models that exhibit a gradual stiffening (e.g. exponential and power-law models), our primary focus is on materials that undergo severe strain-stiffening in the stress-stretch response. In particular, we consider two phenomenological constitutive models that reflect limiting chain extensibility at the molecular level. It is shown that for materials with sufficiently low extensibility no limit point instability occurs and so stable inflation is then predicted for such materials. Potential applications of the results to the biomechanics of soft tissues are indicated.

Peristaltic motion of a power-law fluid in an asymmetric channel

Abstract: The flow of a power-law fluid is investigated in an asymmetric channel caused by the movement of peristaltic waves with the same speed but with different amplitudes and phases on the flexible walls of the channel. The differential equation governing the flow is non-linear and can admit non-unique solutions. There exist two different physically meaningful solutions one satisfying the boundary conditions at the upper wall and the other at the lower wall. The effects of the power-law nature of the fluid on the pumping characteristics and axial velocity are studied in detail.

Nonlinear vibration of symmetrically laminated composite skew plates by finite element method

Abstract: Here, the large amplitude free flexural vibration behavior of symmetrically laminated composite skew plates is investigated using the finite element method. The formulation includes the effects of shear deformation, in-plane and rotary inertia. The geometric non-linearity based on von Karman's assumptions is introduced. The nonlinear matrix amplitude equation obtained by employing Galerkin's method is solved by direct iteration technique. Time history for the nonlinear free vibration of composite skew plate is also obtained using Newmark's time integration technique to examine the accuracy of matrix amplitude equation. The variation of nonlinear frequency ratios with amplitudes is brought out considering different parameters such as skew angle, fiber orientation and boundary condition.

A constitutive model for fibrous tissues considering collagen fiber crimp

Abstract: A micromechanically based constitutive model for fibrous tissues is presented. The model considers the randomly crimped morphology of individual collagen fibers, a morphology typically seen in photomicrographs of tissue samples. It describes the relationship between the fiber endpoints and its arc-length in terms of a measurable quantity, which can be estimated from image data. The collective mechanical behavior of collagen fibers is presented in terms of an explicit expression for the strain-energy function, where a fiber-specific random variable is approximated by a Beta distribution. The model-related stress and elasticity tensors are provided. Two representative numerical examples are analyzed with the aim of demonstrating the peculiar mechanism of the constitutive model and quantifying the effect of parameter changes on the mechanical response. In particular, a fibrous tissue, assumed to be (nearly) incompressible, is subject to a uniaxial extension along the fiber direction, and, separately, to pure shear. It is shown that the fiber crimp model can reproduce several of the expected characteristics of fibrous tissues.

Sequential non-linear least-square estimation for damage identification of structures with unknown inputs and unknown outputs

Abstract: The detection of structural damages real-time on-line, based on vibration data measured from sensors, is an important but challenging research topic, and it has received considerable attentions recently. Due to practical limitations, it is highly desirable to install as few sensors as possible in the structural health monitoring system, leading to incomplete measurements of structural responses and excitations. The traditional time-domain analysis techniques, such as the least-square estimation (LSE) method and the extended Kalman filter (EKF) approach, require that all the external excitations (inputs) be available, which may not be the case for most structural health monitoring systems. Recently, the adaptive sequential non-linear least-square estimate (SNLSE) method has been proposed for the on-line identification of structural damages. In this paper, we extend the SNLSE method to cover the general case with unknown (unmeasured) excitations (inputs) and unknown (unmeasured) acceleration responses (outputs) in order to reduce the number of sensors required in the structural health monitoring system, referred to as the SNLSE-UI-UO. Analytic recursive solutions for the new approach are derived and presented. The accuracy and effectiveness of the proposed approach have been demonstrated using the Phase I ASCE structural health monitoring benchmark building, a 5-degree-of-freedom non-linear hysteretic building model, and a 3-story steel frame finite-element model. Simulation results indicate that the proposed approach is capable of tracking the changes of structural parameters leading to the identification of damages.

Linear and non-linear interactions between static and dynamic bifurcations of damped planar beams

Abstract: The critical and post-critical behavior of a non-conservative non-linear structure, undergoing statical and dynamical bifurcations, is analyzed. The system consists of a purely flexible planar beam, equipped with a lumped visco-elastic device, loaded by a follower force. A unique integro-differential equation of motion in the transversal displacement, with relevant boundary conditions, is derived. Then, the linear stability diagram of the trivial rectilinear configuration is built-up in the parameter space. Particular emphasis is given to the role of the damping on the critical scenario. The occurrence of different mechanisms of instability is highlighted, namely, of divergence, Hopf, double zero, resonant and non-resonant double Hopf, and divergence–Hopf bifurcation. Attention is then focused on the two latter (codimension-two) bifurcations. A multiple scale analysis is carried-out directly on the continuous model, and the relevant non-linear bifurcation equations in the amplitudes of the two interactive modes are derived. The fixed-points of these equations are numerically evaluated as functions of two bifurcation parameters and some equilibrium paths illustrated. Finally, the bifurcation diagrams, illustrating the system behavior around the critical points of the parameter space, are obtained.

Analysis of stagnation point flow toward a stretching sheet

Abstract: There has been much recent interest in the stagnation point flow of a fluid toward a stretching sheet. Investigations that may include oblique stagnation flow and heat transfer to a horizontal plate all involve the same boundary value problem (BVP): f ‴ + ff ″ - ( f ′ ) 2 + b 2 = 0 , f ( 0 ) = 0 , f ′ ( 0 ) = 1 , f ′ ( ∞ ) = b . Here b is the ratio of the stagnation flow strain rate to the stretch rate of the sheet. Through numerical analysis of the problem, several authors have conjectured the existence of a solution for all values of b > 0 . In this note we present numerical evidence that a second solution exists for 0 b b c ≈ 0.16906 . Further we present a mathematical proof that for all b > 0 there exists a monotonic solution to the BVP and if b > 1 , this solution is unique. If b 1 it can be shown that any further solutions cannot be monotonic and the second solution found here numerically is non-monotonic. The asymptotic behavior of solutions near b = 0 and 1 is also presented. Finally, a stability analysis is performed to show that solutions on the upper branch of the dual solutions are linearly stable, while those on the lower branch are linearly unstable.

Destabilization paradox due to breaking the Hamiltonian and reversible symmetry

Abstract: Stability of a linear autonomous non-conservative system in the presence of potential, gyroscopic, dissipative, and non-conservative positional forces is studied. The cases when the non-conservative system is close to a gyroscopic system or to a circulatory one are examined. It is known that marginal stability of gyroscopic and circulatory systems can be destroyed or improved up to asymptotic stability due to action of small non-conservative positional and velocity-dependent forces. The present paper shows that in both cases the boundary of the asymptotic stability domain of the perturbed system possesses singularities such as “Dihedral angle” and “Whitney umbrella” that govern stabilization and destabilization. In case of two degrees of freedom, approximations of the stability boundary near the singularities are found in terms of the invariants of matrices of the system. As an example, the asymptotic stability domain of the modified Maxwell–Bloch equations is investigated with an application to the stability problems of gyroscopic systems with stationary and rotating damping.

Non-linear control of friction-induced self-excited vibration

Abstract: The present paper introduces a new method of controlling friction-driven self-excited vibration. The control law is primarily derived using the Lyapunov's second method. A single degree-of-freedom oscillator on a moving belt represents the primary model of the system. The control action is achieved by modulating the normal load at the frictional interface based on the state of the oscillatory system. The basic mechanism of the control action utilises subcritical Hopf bifurcation of the equilibrium followed by cyclic-fold bifurcation (of limit cycle oscillations) to globally stabilise the equilibrium. The basic mechanism is qualitatively independent of the exact model of friction. Different models of friction, like, algebraic model, LuGre model and switch model with time-dependent static friction are considered to substantiate the above claim. An approximate method for estimating the critical value of the control parameter that ensures global stability of the equilibrium is also proposed.

A novel harmonic balance analysis for the Van Der Pol oscillator

Abstract: This study focuses on a novel harmonic balance formulation, the high-dimensional harmonic balance method. To investigate a non-linearity in the damping term, the system chosen for study is the Van der Pol's oscillator. Both unforced and forced oscillators are analyzed. The results from the analysis are compared with those obtained from the classical harmonic balance and the time marching (Runge–Kutta) methods.

Physical mechanism of the destabilizing effect of damping in continuous non-conservative dissipative systems

Abstract: The paper attempts to give a physical explanation of the mechanism behind the so-called destabilizing effect of small internal damping in the dynamic stability of Beck's column. Both internal (material) and external (viscous fluid) damping are considered. An energy equation is derived for the balance between the work done by the non-conservative ‘follower force’ and the energy dissipated by the internal and external damping forces. Evaluated at the critical load, where a flutter instability is initiated, this equation explicitly shows the influence of damping upon flutter frequency, phase angle, and vibration amplitude. The gradient of the phase angle, evaluated at the free end of the column, is found to be the ‘valve’, which controls how much work the follower force can do on the column during each period of oscillation. And a large change in this gradient with increasing—but still small—internal damping is found to be responsible for the destabilizing effect.

Thin-plate theory for large elastic deformations

Abstract: Non-linear plate theory for thin prismatic elastic bodies is obtained by estimating the total three-dimensional strain energy generated in response to a given deformation in terms of the small plate thickness. The Euler equations for the estimate of the energy are regarded as the equilibrium equations for the thin plate. Included among them are algebraic formulae connecting the gradients of the midsurface deformation to the through-thickness derivatives of the three-dimensional deformation. These are solvable provided that the three-dimensional strain energy is strongly elliptic at equilibrium. This framework yields restrictions of the Kirchhoff–Love type that are usually imposed as constraints in alternative formulations. In the present approach they emerge as consequences of the stationarity of the energy without the need for any a priori restrictions on the three-dimensional deformation apart from a certain degree of differentiability in the direction normal to the plate.

A finite element study of the deformations, forces, stress formations, and energy losses in sliding cylindrical contacts

Abstract: This work presents the results of a finite element analysis (FEA) used to simulate two-dimensional (2D) sliding between two interfering elasto-plastic cylinders The material for the cylinders is modeled as elastic-perfectly plastic and follows the von Mises yield criterion The FEA provides trends in the deformations, reaction forces, stresses, and net energy losses as a function of the interference and sliding distance between the cylinders Results are presented for both frictionless and frictional sliding and comparisons are drawn The effects of plasticity and friction on energy loss during sliding are isolated This work also presents empirical equations thatt relate the net energy loss due to sliding under an elasto-plastic deformation as a function of the sliding distance Contour plots of the von Mises stresses are presented to show the formation and distribution of stresses with increasing plastic deformation as sliding progresses This work shows that for the plastic loading cases the ratio of the horizontal force to the vertical reaction force is non-zero at the point where the cylinders are perfectly aligned about the vertical axis In addition, a “load ratio” of the horizontal tugging force to the vertical reaction force is defined Although this is analogous to the common definition of the coefficient of friction between sliding surfaces, it just contains the effect of energy loss in plasticity The values of the contact half-width are obtained for different vertical interferences as sliding progresses

Near-grazing Dynamics in Tapping-mode Atomic-force Microscopy

Abstract: In tapping-mode atomic-force microscopy, non-linear effects due to large variations in the force field on the probe tip over very small length scales and the intermittency of contact may induce strong dynamical instabilities. In this paper, a discontinuity-mapping-based analysis is employed to investigate the destabilizing effects of low-velocity contact on a lumped-mass model of an oscillating atomic-force-microscope cantilever tip interacting with a typical sample surface. As illustrated using two tip–sample force models, the analysis qualitatively captures the potential loss of stability and disappearance of a low-contact-velocity steady-state response. The quantitative agreement of the predictions of the discontinuity-mapping-based analysis with direct numerical simulations, at least for sufficiently low contact velocity, supports its use in the passive redesign or active control of the tip–sample mechanism for purposes of preventing such a loss of stability.