Vibratory characteristics of flexural non-uniform Euler–Bernoulli beams carrying an arbitrary number of spring–mass systems

A new exact method for the analysis of free flexural vibrations of non-uniform multi-step Euler–Bernoulli beams carrying an arbitrary number of single-degree-of-freedom and two-degree-of-freedom spring–mass systems is presented in this paper. The closed-form solutions for free vibrations of non-uniform Euler–Bernoulli beams are derived for five important cases. Then, using the massless equivalent springs to replace the spring–mass systems and the fundamental solutions developed in this paper, the frequency equation for free flexural vibrations of a multi-step non-uniform beam with any kind of support configurations and carrying an arbitrary number of spring–mass systems can be conveniently established from a second-order determinant. The proposed method is computationally efficient due to the significant decrease in the determinant order as compared with previously developed procedures.     

Erosion–corrosion degradation mechanisms of Fe–Cr–C and WC–Fe–Cr–C PTA overlays in concentrated slurries

In this investigation the microstructure and erosion–corrosion behaviour of a Fe–Cr–C overlay (FeCrC–matrix) produced by plasma transferred arc welding (PTA) and its metal matrix composite (FeCrC–MMC) were assessed. The FeCrC–MMC was obtained by the addition of 65 wt.% of tungsten carbide (WC). The erosion–corrosion tests (ECTs) were carried out using a submerged impinging jet (SIJ); after the ECTs the surface of the overlays was analysed to identify the damage mechanisms. Two different temperatures (20 and 65 °C) and sand concentrations (10 and 50 g/l) were used in a solution of 1000 ppm of Cl⁻ and a pH value of 8.5; the conditions were chosen to be representative of the recycling water in the tailings line in the oilsands industry. The FeCrC–matrix showed a dendritic structure and a high concentration of carbides in the interdendritic zone. The addition of the WC reinforcing phase promoted the formation of W-rich intermetallic phases, increased the microhardness values of the matrix phase of the FeCrC–MMC overlay and dramatically improved its erosion–corrosion performance as expected. For the FeCrC–matrix overlay the main erosion–corrosion degradation mechanisms were severe plastic deformation and the formation and removal of material flakes due to consecutive impacts. At 65 °C the dendritic zone was severely corroded in the area of low impact frequency. The FeCrC–MMC showed greater attack of the matrix phase compared to the WC grains; at high sand concentration the WC grains were severely fractured and flattened. The anodic polarisation analysis showed active corrosion behaviour of the FeCrC–MMC at both temperatures and sand concentrations; however the temperature dramatically increased the corrosion process of the surface studied under erosion–corrosion conditions. The paper assesses the degradation mechanisms of both FeCrC–matrix and FeCrC–MMC with the aim of understanding what aspects of MMCs must be adapted for optimum erosion–corrosion resistance.     

Stochastic bending–torsion coupled response of axially loaded slender composite-thin-walled beams with closed cross-sections

The stochastic bending–torsion coupled response of axially loaded slender composite beams with solid or thin-walled closed cross-sections are investigated by using normal mode method in conjunction with receptance method. The classical composite beam theory with shear deformation and rotary inertia ignored is employed and the effects of bending–torsion coupling and axial force are included in the present formulations. The theoretical expressions for the displacement response of axially loaded slender composite beams subjected to concentrated or distributed stochastic excitations with stationary and ergodic properties are derived. The proposed method is illustrated by its application to two particular examples to study the effects of bending–torsion coupling and axial force on the stochastic response of the composite beams.     

Improved flexural–torsional stability analysis of thin-walled composite beam and exact stiffness matrix

A simple but efficient method to evaluate the exact element stiffness matrix is newly presented in order to perform the spatially coupled stability analysis of thin-walled composite beams with symmetric and arbitrary laminations subjected to a compressive force. For this, the general bifurcation-type buckling theory of thin-walled composite beam is developed based on the energy functional, which is consistently obtained corresponding to semitangential rotations and semitangential moments. A numerical procedure is proposed by deriving a generalized eigenvalue problem associated with 14 displacement parameters, which produces both complex eigenvalues and multiple zero eigenvalues. Then the exact displacement functions are constructed by combining eigenvectors and polynomial solutions corresponding to non-zero and zero eigenvalues, respectively. Consequently exact element stiffness matrices are evaluated by applying member force–displacement relationships to these displacement functions. As a special case, the analytical solutions for buckling loads of unidirectional and cross-ply laminated composite beams with various boundary conditions are derived. Finally, the finite element procedure based on Hermitian interpolation polynomial is developed. In order to verify the accuracy and validity of this study, the numerical, analytical, and the finite element solutions using the Hermitian beam elements are presented and compared with those from ABAQUS's shell elements. The effects of fiber orientation and the Wagner effect on the coupled buckling loads are also investigated intensively.     

Safety and collapse of elastic–plastic beams against dynamic loads

The dynamic load-bearing capacity of elastic–plastic beam structures is analysed by the apparatus of shakedown theory. The reduced kinematic formulation for bending beams, which is equivalently deduced from Koiter’s kinematic theorem, combined with the plastic collapse’s method of hinge mechanisms appears effective in solving practical problems. The safety limits on the quasiperiodic dynamic loads as well as respective collapse mechanisms for a number of practical beams are determined.     

Nanoindentation and nanowear of extremely thin protective layers of C–N and B–C–N

Carbon nitride (C–N) and boron and carbon nitride (B–C–N) films, 1 and 3 nm thick, were deposited on magnetic disks by means of a complex treatment method involving plasma irradiation and CN or BCN reactive sputtering in nitrogen and helium mixed gas using two targets of h-BN and graphite. The properties of these extremely thin coatings were evaluated by indentation using an atomic force microscope and nanowear tests using a lateral modulation friction force microscope. The extremely thin B–C–N coatings show highest indentation hardness and good wear resistance properties. It is proposed that this is due to their graduated composition and interfacial properties.     

Dynamic analysis of smart composite beams by using the frequency-domain spectral element method

To excite or measure the dynamic responses of a laminated composite structure for the active controls of vibrations or noises, wafer-type piezoelectric transducers are often bonded on the surface of the composite structure to form a multi-layer smart composite structure. Thus, for such smart composite structures, it is very important to develop and use a very reliable mathematical and/or computational model for predicting accurate dynamic characteristics. In this paper, the axial-bending coupled equations of motion and boundary conditions are derived for two-layer smart composite beams by using the Hamilton??s principle with Lagrange multipliers. The spectral element model is then formulated in the frequency domain by using the variation approach. Through some numerical examples, the extremely high accuracy of the present spectral element model is verified by comparing with the solutions by the conventional finite element model provided in this paper. The effects of the lay-up of composite laminates and surface-bonded wafer-type piezoelectric (PZT) layer on the dynamics and wave characteristics of smart composite beams are investigated. The effective constraint forces at the interface between the base beam and PZT layer are also investigated via Lagrange multipliers.     

Reproducing hoop stress–strain behavior for tubular material using lateral compression test

Identification of material properties in the hoop direction, such as stress–strain behavior, is essential in tube hydroforming processes. Conventional tests such as uniaxial tension and compression tests have some drawbacks and limitations. In the current investigations a simple technique to identify the stress–strain behavior in the hoop direction for tubular material is introduced, based on the experimental data obtained from tube lateral compression test. In the proposed technique, an assumed stress–strain curve is used in finite element simulation to predict the load deflection curve of the tube lateral compression. An iterative algorithm is used to compare the calculated and experimental load deflection curves until a good agreement with a percentage deviation less than 4% is obtained. The suggested technique was used to obtain the material properties of Cu–40%Zn brass tube. The predicted stress–strain curve was compared with that obtained from uniaxial compression test. Comparison between the experimental and predicted stress–strain curve showed that the proposed technique is effective in the prediction of the material properties from the tube lateral compression test with percentage deviation less than 1%.     

Dry wear and friction behaviour of plasma nitrided Ti–6AL–4 V alloy after explosive shock treatment

The unlubricated wear behaviour of explosive shock treated and, subsequently plasma nitrided Ti–6Al–4 V alloy was studied using a ball-on-disc wear tester. Plasma nitriding was carried out at three different temperatures (700, 800 and 900 °C) for 3, 6, 9 and 12 h. Plasma nitriding after explosive shock treatment enabled a reduction in the wear rate of two orders of magnitude. Detailed investigations of this improved wear performance dependent on the nitriding temperature and time were carried out. The friction and wear data showed a clear breakthrough transition from the nitrided layer to the core of the Ti–6Al–4 V alloy matrix. The lowest wear volume was obtained for the sample, nitrided at 900 °C for 12 h, especially at loads of 2.5, 5 and 7.5 N. Obviously, the hard nitride layers were intimately associated with low wear rate, providing a smooth low friction surface. The coefficient of friction reduced from 0.46 to 0.2 due to a thick and hard compound layer resulting from a high nitrogen diffusion rate caused by explosive shock treatment that expected to increase point defects in the alloy. Detailed examination of the wear tracks showed that plasma nitriding changes the mechanism of wear from one of adhesion for untreated Ti–6Al–4 V to both delamination and mild abrasive.     

Adaptive Vold–Kalman filtering order tracking

This study proposes and implements an adaptive Vold–Kalman filtering order tracking (VKF_OT) approach to overcome the drawbacks of the original VKF_OT scheme for condition monitoring and diagnosis of rotary machinery. The paper comprises theoretical derivation and numerical implementation. Comparisons of the adaptive VKF_OT scheme to the original are accomplished through processing two synthetic signals composed of close orders and crossing orders, respectively. Parameters such as the weighting factor and the correlation matrix of process noise, which influence tracking performance, are investigated. The adaptive scheme coping with end effects of computation can simultaneously extract multiple order/spectral components, and effectively decouple close and/or crossing orders associated with multi-axial reference rotating speeds. Furthermore, the adaptive OT scheme is realized through Kalman filtering based upon adapted one-step prediction scheme, where a parameter, the weighting factor, is newly introduced into the computation. Thus the technique can be computed on-line and implemented as a real-time processing application.     

Improvement of the wear behavior of stircast Al–Si–Pb alloys by hot extrusion

The wear behavior of as-cast and hot extruded Al–Si–Pb alloys were investigated under dry conditions using a pin-on-disc type wear testing machine. The results show that the microstructure and mechanical properties can be greatly improved and porosity can be significantly decreased by hot extrusion. These factors contribute to great increase in wear resistance of hot extruded Al–Si–Pb alloys. Optical observation and X-ray photoelectron spectroscopy (XPS) analysis reveal the almost constant wear rate at mediate load levels. Better resistance to seizure for Al–Si–Pb alloys with more than 15 wt% lead are due to a film of lubricant covering almost the entire worn surface. This film is a mixture of different constituents containing Al, Fe, Si, O and Pb.     

Investigation of sharp contact at rigid–plastic conditions

Sharp contact problems are examined theoretically and numerically. The analysis is focused on elastic–plastic material behaviour and in particular the case when the local plastic zone arising at contact is so large that elastic effects on the mean contact pressure will be small or negligible. It is shown that, save for the particular case of a rigid–plastic power-law material, at such conditions, there is no single representative value on the uniaxial stress-strain curve that can be used in order to evaluate the global parameters at contact. However, the present numerical results indicate that good accuracy predictions for the mean contact pressure can be achieved when this variable is described by two parameters corresponding to the stress levels at, approximately, 2 and 35% plastic strain. Regarding the size of the contact area, it is shown that this quantity is very sensitive to elastic effects and any general correlation with material properties is complicated at best. The numerical analysis is performed by using the finite element method and the theoretical as well as the numerical results are compared with relevant experimental ones taken from the literature. From a practical point of view, the presented results are directly applicable to material characterization or measurements of residual mechanical fields by sharp indentation tests, but also for situations such as contact in gears or in electronic devices.     

Analysis of viscosity interaction on the misaligned conical–cylindrical bearing

Viscosity is an essential property in hydrodynamic lubrication. In general, the lubricant is not considered to have uniform viscosity within a given bearing. The viscosity of the lubricant is affected by both pressure and temperature. The viscosity of the lubricant increases with pressure and for most lubricants, this effect is much larger than that of temperature or shear when the pressure is significantly above than the atmospheric pressure. This study analyzes the thermal effect of conical–cylindrical bearing performance parameters via the viscosity–pressure–temperature relationships of lubricants. The results reveal that pressure increases both the film viscosity and temperature as well.     

Modeling and estimation of deformation behavior of particle-reinforced metal–matrix composite

In order to estimate the characteristic feature of the deformation behavior of materials with a length scale, the strain gradient plasticity theories, corresponding variational principle and a finite element method are given. Then the finite element method is applied to the estimation of the mechanical characteristics of the particle reinforced metal–matrix composites modeled under plane strain conditions. The effects of the volume fraction, size and distribution pattern of the reinforcement particles on the macroscopic mechanical property of the composite are discussed. It has been clarified that the deformation resistance of the composite is substantially increased with decreasing particle size under a constant volume fraction of the reinforcement material. The main cause of the increase of the deformation resistance in the plastic range is the high strain gradient appearing in the matrix material, which increases with the reduction of the distance between particles.     

An efficient numerical method for predicting the natural frequencies of cylindrical helical springs

In this study, the stiffness method is employed for the free vibration problem of cylindrical helical springs. The element stiffness matrix for the helical spring with twelve degrees-of-freedom is obtained exactly by the transfer matrix method. The efficacious numerical algorithm is employed for the computation of the element transfer matrix. The concentrated element mass matrix is used. The subspace iteration method is preferred for the solution of the large-scale eigenvalue problem. The axial and shear deformation and the rotary inertia terms are considered in the formulation. The free vibrational parameters are chosen as the number of coils (n=3–16), the helix pitch angle (α=5–25°), the shape of cross-section (circular, hollow circle and squared) and as the ratio of the diameters of cylinder to wire (D/d=4–16) in a wide range. Solving the miscellaneous problems, the non-dimensional charts are obtained for the cylindrical helical springs fixed at both ends. Using these charts the natural frequencies are expressed in analytical form in a very good approximation (with the maximum absolute relative error of 5%) and presented for the designers.     

Mechanical and microstructural analysis on the superplastic deformation behavior of Ti–6Al–4V Alloy

The present investigation has been made to study the superplastic deformation behavior of Ti–6Al–4V alloy based on the theory of inelastic deformation, and to analyze the boundary sliding characteristics using transmission electron microscopy. Flow characteristics for the microstructures of 2.5–16 μm grain sizes were analyzed by the load relaxation tests at various temperatures ranging from 600 to 927°C. The results showed that at relatively low temperatures such as 600°C the grain matrix deformation was dominant and found to be consistent with the state equation based on the dislocation dynamics. On the contrary, above the temperature of 800°C, the grain boundary sliding became dominant resulting in the change of curvature in the stress–strain rate curves, which was more pronounced in the finer microstructures. However, the deformation mode changes from grain boundary sliding to grain matrix deformation with the increase in grain size as evidenced by transmission electron microscopy.     

Thermal stability and high-temperature wear of Ti–TiN and TiN–CrN nanomultilayer coatings under self-mated conditions

Ti–TiN and TiN–CrN nanomultilayers were thermally stable retaining uniform and sharp layer interfaces up to 24 h at 773 K, without any oxidation or phase transformation accompanying each individual layer. Decreasing the multilayer spacing resulted in an increase in the hardness in both cases. The coating hardness was found to be independent of the substrate type, when applied on HS718, Ti64 and HCHCr substrates. In scratch testing, the multilayers displayed a better resistance to the onset of failure, as compared to the monolayer TiN. The substrate plasticity played an important role in determining the coating failure mode. Self-mated wear tests revealed the CrN–TiN system to exhibit the best wear behaviour, both at room temperature and at 773 K. The Ti–TiN coatings are more accommodative with all three substrates, as compared to TiN–CrN and TiN.     

A further study about the behaviour of foundation piles and beams in a Winkler–Pasternak soil

The natural vibrations and critical loads of foundation beams embedded in a soil simulated with two elastic parameters through the Winkler–Pasternak (WP) model are analysed. General end supports of the beam are considered by introducing elastic constraints to transversal displacements and rotations. The solution is tackled by means of a direct variational methodology previously developed by the authors who named it as whole element method. The solution is stated by means of extended trigonometric series. This method gives rise to theoretically exact natural frequencies and critical loads. A particular behaviour arises from the analysis of the lateral soil influence. It is found that the boundary conditions of the beam are influenced by the soil at the left and right sides of the beam. The possible alternatives are that the soil be cut or dragged by the non-fixed ends of the beam. In the standard WP model, the lateral soil influence is not considered. Natural frequencies and critical load numerical values are reported for beams and piles elastically supported and for various soil parameters. The results are found with arbitrary precision depending on the number of terms taken in the series. Some unexpected modes and eigenvalues are found when the different alternatives are studied. It should be noted that this special behaviour is present only when the Pasternak contribution is taken into account.     

Erosion– and cavitation–corrosion of titanium and its alloys

The economic and effective operation of machinery and plant involved in fluids handling is increasingly dependent on the utilisation of materials that combine high corrosion resistance and good wear resistance. This paper studies two wear–corrosion situations: (1) erosion–corrosion, where the wear is due to impacting solids in a liquid medium and (2) cavitation–corrosion, where the wear is due to impacting liquid micro-jets formed by imploding air bubbles. The characteristics of a commercially pure titanium (CP-Ti) and three alloys in erosion–corrosion and cavitation–corrosion conditions have been studied. The erosion–corrosion characteristics of each material was assessed using an impinging-jet apparatus. The tests were performed at an angle of impingement of 90°C at a particle velocity of 17 m/s and in a saline solution of 3.5% NaCl at 18°C. A series of experiments was conducted to determine the mass loss by combined erosion–corrosion before independently determining the electrochemical corrosion contribution to mass loss. It has been shown that exposure to liquid–solid erosion causes disruption of the passive film on Ti and active corrosion occurs. In contrast, the materials exhibited passive behaviour in static conditions and when exposed to a cavitating liquid only CP-Ti became active. The role of corrosion in these wear–corrosion environments on CP-Ti and Ti-alloys is discussed in this paper.