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.     

Dissimilar ultrasonic spot welding of Mg-Al and Mg-high strength low alloy steel

Sound dissimilar lap joints were achieved via ultrasonic spot welding (USW), which is a solid-state joining technique. The addition of Sn interlayer during USW effectively blocked the formation of brittle al₁₂Mg₁₇ intermetallic compound in the Mg-Al dissimilar joints without interlayer, and led to the presence of a distinctive composite-like Sn and Mg₂Sn eutectic structure in both Mg-Al and Mg-high strength low alloy (HSLA) steel joints. The lap shear strength of both types of dissimilar joints with a Sn interlayer was significantly higher than that of the corresponding dissimilar joints without interlayer. Failure during the tensile lap shear tests occurred mainly in the mode of cohesive failure in the Mg-Al dissimilar joints and in the mode of partial cohesive failure and partial nugget pull-out in the Mg-HSLA steel dissimilar joints.     

Transient longitudinal strain changes during welding

Experimental measurements are presented to demonstrate the longitudinal strain changes that occur when structural sections are welded. The measurements show that the final deformation in the upset zone can be as much as 35 percent more than the transient deformation on heating. The assumption in the Okerblom analysis that these two deformations are equal is discussed in view of the results.¹ In addition, measured final deformations are compared to those predicted from this same analysis for a wide range of edge-welded rectangular bar sections.     

Buckling of reinforced fibers during explosive welding

This paper is devoted to the study of processes of buckling of fibers while fabricating fiber-reinforced composite materials by explosive welding. The process of welding of metals by itself is not studied in this article. Up to the moment of explosion, the multi-layered workpieces consist of alternate layers of plane sheets and parallel bars of rein-forcement. Matrix is fabricated as a result of welding of plane sheets. In the course of collision at explosive welding, the reinforcing bars are subjected to intensive longitudinal impact. The modes of buckling of reinforcement obtained in a theoretical analysis are compared with experimental modes. It has been established that, to describe satisfactorily the experimental results, in a theoretical analysis one must take into account the interaction of longitudinal and transversal oscillations of bars.     

A model of the thermomechanical process during welding of plates

A model of the thermomechanical process that takes place as straight plates are welded, with due consideration of the weld seam filling, is developed within the limits of linear thermoelasticity from the temperature dependence of the properties. The filler material is assumed to be unstressed as the weld seam is filled. A modified form of constitutive equations for the filler material is proposed for implementing that assumption within the standard finite-element technique. Some laws of relative plate displacement during welding are investigated and experimental and computed results are compared. Translated from Prikladnaya Mekhanika, Vol. 34, No. 12, pp. 70–76, December, 1998.     

Determining the contact force during the transverse impact of plates

The contact force during the transverse impact of a plate is determined from dynamic strain-gage measurements made on the plate. Experimental results for the impact of an aluminum plate are presented, and comparisons are made with finite-element predictions and measurements from a force transducer. Paper was presented at the 1986 Spring Conference on Experimental Mechanics held in New Orleans, LA on June 8–13.     

The effect of strain rate and heat developed during deformation on the stress-strain curve of plastics

Polymethylmethacrylate, cellulose acetate butyrate, polypropylene and nylon 6–6 have been characterized in compression at various strain rates from 10^?4 s^?1 to 10³ s^?1 at room temperature. A medium strain-rate machine and a split-Hopkinson-bar apparatus are used in conducting the experiments. The temperature rise developed during deformation is also measured by using a thermocouple. All four materials tested definitely show a viscous effect at the beginning of the deformation and a plastic flow follows thereafter. Test results also indicate that the temperature rise developed during deformation cannot be neglected in determining the dynamic response of those materials investigated in this study.     

The effect of tangential elasticity of the contact layer between stator and rotor in travelling wave ultrasonic motors

In travelling wave ultrasonic motors the elliptical motion of material points of the stator drives the rotor due to frictional mechanisms. The motor characteristic strongly depends on the mechanical properties of the components stator, rotor and contact layer. In order to predict the motor behaviour, a model for the contact between stator and rotor has been developed. The goal of the present paper is to point out the importance of the tangential elasticity of the contact layer which is responsible for the formation of stick zones and also for the amount of friction losses and overall efficiency. Therefore a comparison with a model with a contact layer rigid in tangential direction is given. Based on a visco-elastic foundation model for the contact layer, torque-speed curves as well as torque-efficiency curves are computed. Experimental investigations for identification of parameters, check of assumptions and model validation are carried out. Finally, the model is used to show the results of parameter variations for normal force, vibration amplitude and modulus of elasticity of the contact layer.     

Surface versus bulk nucleation of dislocations during contact

The indentation of single crystals by a periodic array of flat rigid contacts is analyzed using discrete dislocation plasticity. Plane strain analyses are carried out with the dislocations all of edge character and modeled as line singularities in a linear elastic solid. The limiting cases of frictionless and perfectly sticking contacts are considered. The effects of contact size, dislocation source density, and dislocation obstacle density and strength on the evolution of the mean indentation pressure are explored, but the main focus is on contrasting the response of crystals having dislocation sources on the surface with that of crystals having dislocation sources in the bulk. When there are only bulk sources, the mean contact pressure for sufficiently large contacts is independent of the friction condition, whereas for sufficiently small contact sizes, there is a significant dependence on the friction condition. When there are only surface dislocation sources the mean contact pressure increases much more rapidly with indentation depth than when bulk sources are present and the mean contact pressure is very sensitive to the strength of the obstacles to dislocation glide. Also, on unloading a layer of tensile residual stress develops when surface dislocation sources dominate.     

Influence of calcium on cracking of steel during the welding or casting process

The influence of calcium on the brittle–ductile transition temperature is studied for the solidification range. The test bars were cast from carbon steel with and without the addition of calcium. Mechanical properties at different temperatures around the solidus were evaluated. The fractured surfaces were examined using a scanning microscope coupled with a Kevex analyser. For the modified steel, the liquid phase appears (on heating) for higher temperature while the temperature range within which fracture can nucleate, during casting or welding, is markedly narrowed. In medium-carbon steel, calcium changes the manner of fusion of the grains. The inclusions of the liquid phase form at the triple points and the surfaces of grains remain solid above the solidus temperature.     

Injection moulding of plastics

Summary In continuation of a previous investigation a simple analytical expression is derived in closed form for the thickness distribution of the freeze-off layer which is vitrified at the (flat) wall of an oblong rectangular cavity. As has been pointed out previously, this layer is marked for amorphous polymers by the molecular orientation (birefringence pattern) in the moulded sample. One can show that a more detailed study with the aid of the coupled equations of energy and of motion will not furnish essential improvements. Problems of polymer physics like glass transition or crystallization kinetics at extreme rates of cooling and shearing must be solved first.

---

Zusammenfassung In Fortsetzung einer früheren Untersuchung wurde ein einfacher analytischer Ausdruck in geschlossener Form für die Dickenverteilung der eingefrorenen Schicht abgeleitet, die an der (flachen) Wand eines langgestreckten rechteckigen Formnestes während des Einspritzvorgangs glasig erstarrt. Wie früher auseinandergesetzt wurde, wird diese Schicht bei amorphen Polymeren durch die Molekülorientierung (Doppelbrechungsmuster) im gespritzten Formteil markiert. Man kann zeigen, daß eine eingehendere Studie mit Hilfe der gekoppelten Energie- und Impulsgleichungen keine essentiellen Verbesserungen bringt. Probleme der Polymerphysik, wie Glasübergang oder Kristallisationskinetik bei extremen Abkühlungs- und Schergeschwindigkeiten, müssen erst gelöst werden.

---

List of Symbols a heat diffusivity of polymer melt (averaged overT) [m²s^–1] - B breadth of mould cavity [m] - Br Brinkman number ( ) - c heat capacity of polymer melt (averaged overT) [J kg^–1 K^–1] - F ₀ Fourier number (at _i/4H ²) - h heat transfer coefficient by melt flow [J K^–1 s^–1 m^–2] - h heat transfer coefficient by layer growth [J K^–1 s^–1 m^–2] - H half height of mould cavity [m] - L length of mould cavity [m] - n exponent in eq. [18] (= 0.417) - Nu Nußelt number (2Hh/) - P pressure gradientdP/dz in mould [N m^–3] - t time [s] - t _i injection time [s] - T _g glass transition temperature of polymer [K] - T _i injection temperature of polymer melt [K] - T _l stagnation temperature [K] - T _m mould wall temperature [K] - speed of flow front during mould filling [m s^–1] - x coordinate perpendicular to mould wall [m] - z coordinate in the injection direction [m] - thickness of stagnant layer (atT _l) [m] - ₀ optically detectable freeze-off thickness [m] - ⁺ apparent layer thickness (atT _i) [m] - dimensionless freeze-off thickness (= ₀/2H) - dimensionless distance from entrance (=z/L) - _m dimensionless coordinate of layer maximum - _g dimensionless temperature (= (T _i –T _l)/(T _g –T _m)) - _i dimensionless temperature (= (T _i –T _l)/(T _i –T _m)) - _l dimensionless temperature (= (T _i –T _l)/(T _l –T _m)) - _i viscosity of polymer atT _i [N s m^–3] - _l viscosity of polymer atT _l [N s m^–3] - heat conductivity of polymer melt (averaged) [J K^–1 s^–1 m^–1] - density of polymer melt (averaged) [kg m^–3] - dimensionless time (eq. [11]) - ⁺ dimensionless parameter (eqs. [19a] and [19b]) - dimensionless layer thickness (eq. [12]) - ⁺ dimensionless parameter (eq. [20a]) - dimensionless parameter (eqs. [11a] and [11b]) Formerly at Delft University of Technology, Delft (The Netherlands).Paper presented at the Conference on Chemical Engineering Rheology, Annual Meeting of the Deutsche Rheologische Gesellschaft in Aachen, March 5–7, 1979.With 3 figures and 1 table     

Optimization of travelling wave ultrasonic motors using a three-dimensional analysis of the contact mechanism at the stator–rotor interface

The present paper investigates optimization rules and new design methodologies dealing with the contact mechanics in rotative travelling wave ultrasonic motors (TWUM). The proposed approaches focus on the design of the rotor, including the friction layer that is usually deposited onto its lower surface, while the stator is supposed to be preliminary designed. Contact aspects such as the transmission of mechanical power as well as the wear mechanism of the friction layer are investigated, according to the analysis of the stator–rotor contact mechanics in both hoop and radial directions. Considering a classical wear criterion in a preliminary step, a contact ratio, that allows the mechanical power to be optimized, is pointed out in the hoop direction. In a further step, the contact conditions in the radial direction are improved through the elastic fitting of the stator and rotor radial deflexions, therefore allowing the material's wear to be decreased. Some experimental tests, that have been recently performed, give a comparison of wear marks, which occur onto optimal and non-optimal rotor geometries. A first mechanism synthesis is finally proposed in such a way to allow the mechanical architecture of the rotor (including the friction layer) to be automatically designed according to a given set of mechanical constraints.     

Kinetics of the collision of metallic plates in a multilayer packet during explosion welding

Volgograd. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 5, pp. 173–175, September–October, 1994     

Computational mechanics modelling of cell-substrate contact during cyclic substrate deformation

Experimental testing carried out on various adherent cell types cultured on deformable substrates reveals specific patterns of cell reorientation in response to cyclic stretching of the substrate. In Wang et al. (2001. Specificity of endothelial cell reorientation in response to cyclic mechanical stretching. J. Biomech. 34, 1563), a number of substrate deformation modes were considered: in cases where lateral deformation of the substrate was prohibited (uniaxial case) cells were found to elongate perpendicular to the stretch direction, whereas in cases where the substrate was laterally unrestrained (biaxial case) cells were found to elongate at an angle to the stretch direction. The alignment directions in both cases corresponded to directions of minimum substrate strain. However, the mechanisms underlying such behaviour are not apparent from such in-vitro testing and consequently are not well understood. In this study finite element models are developed in order to investigate the role of cell viscoelasticity in cell debonding and cell realignment under conditions of cyclic substrate stretching using cohesive zone formulations to simulate cell-substrate interfacial behaviour. The characteristic length scale used in such models is based on the length of the receptor-ligand bonds at the cell-substrate interface. Two-dimensional simulations reveal that permanent debonding at the cell-substrate interface occurs due to the accumulation of strain concentrations in the cell. Inclusion of a nucleus in two-dimensional models is shown to have little effect on debonding while discrete cell-substrate contact at focal adhesion sites results in a completion of debonding in fewer cycles. Three-dimensional cohesive zone models are developed in order to compute changes in cell-substrate contact under the aforementioned uniaxial and biaxial modes of substrate deformation. Results reveal that, due to the accumulation of tensile and compressive strains in the cell under cyclic deformation, definite patterns of cell-substrate contact area evolution are computed. With continued cycling, equilibrium contact area profiles with definite orientations are established. These orientations are found to be coincidental with the preferential cell orientation directions seen in the experiments. As no changes in cell morphology are predicted by the models it is concluded that permanent breaking of cell-substrate bonds constitutes the first stage in the process of cell alignment under such mechanical loading.