Displacement Equations Research Articles

The effect of imperfect bonding on stress distribution in fibrous composites

An attempt was made to map the distribution of stress in fibrous composites with imperfect bonding. Two analytical micro-mechanics models were developed. In the first model, the composite was subjected to axial tensile loading, parallel to the fiber direction, and the assumption of iso-strain was employed to derive the control equations. In the second model, the composite was loaded in the direction transverse to the fiber. An iso-stress condition was employed, and Airy stress function was utilized to articulate the stress and displacement equations. An assumption of how the stress is transferred between the matrix and the fiber was introduced in both models. To investigate and validate the models, specimens were fabricated using a carbon plain weave fabric and a geopolymer matrix. Single fiber pullout and three-point bending tests were carried out. The maximum average tensile stress obtained from the three-point bending tests, as well as the mechanical properties of the fiber and geopolymer, served as input for the models. Results indicate that the effect of the level of bonding is very high in the transverse direction while almost negligible in the axial direction. The difference in the maximum value of the axial tensile stress at the fiber-matrix interface was used to calculate the numerical value of the interfacial shear strength, and the numerical result matched the data obtained from the single fiber pullout test.

Optimal stability for laminated beams with Kelvin–Voigt damping and Fourier’s law

This article deals with the asymptotic behavior of a mathematical model for laminated beams with Kelvin–Voigt dissipation acting on the equations of transverse displacement and dimensionless slip. We prove that the evolution semigroup is exponentially stable if the damping is effective in the two equations of the model. Otherwise, we prove that the semigroup is polynomially stable and find the optimal decay rate when damping is effective only in the slip equation. Our stability approach is based on the Gearhart–Prüss–Huang Theorem, which characterizes exponential stability, while the polynomial decay rate is obtained using the Borichev and Tomilov Theorem.

Dynamic model for an independent suspension

The paper presents the dynamic model with 4 degrees of freedom of an independent suspension with MacPherson type coil springs, taking into account the anti-roll bar. Starting from the constructive scheme, a kinematic analysis necessary to obtain the dynamic model is carried out. The differential displacement equations are obtained starting from the expressions of kinetic and potential energy, by applying Lagrange’s equations. Next, the expressions of the inertia, stiffness and damping matrices and, finally, the matrix expression of pitch-roll vibrations are presented. At the end of the paper, in the numerical application carried out with a calculation program in MATLAB, the influence of the values of the damping constants on the dynamic forces at the wheels is studied. The numerical results are presented in the form of diagrams.

A semi-analytical method for stress-strain analysis of tunnels under dip-slip faulting based on an elastic-plastic foundation beam model

Tunnels under faulting will be seriously damaged, and an accurate analytical method is an effective tool for tunnel response assessment in preliminary design. A series of mechanical analytical methods have been developed for tunnels under faulting. However, some unrealistic assumptions are still employed in the analytical method for tunnels under dip-slip faulting, including tunnel segmentation, constant stratum stress, and axial force in governing equations of tunnel displacement, which increases the solution complexity and decreases the accuracy. In this paper, a tunnel crossing an active fault is simplified as a large deformed continuous beam acting on an elastic-plastic foundation, and the tunnel-stratum interaction is treated as a series of axial and vertical elastic-plastic springs; the elastic-plastic tunnel-stratum interaction stresses and axial force are considered in the governing equations by using the finite difference method and a novel iterative method. In this way, the three assumptions employed in the existing analytical methods are removed. Later, the proposed method is verified by comparing it with the results from the model test, numerical simulation, and existing method. The comparisons show that the results of the proposed method agree excellently with those from the model test and numerical model, and the accuracy is higher than that of the existing analytical method. Finally, the failure scope and stress distribution of tunnels under the action of normal and reverse faulting are analyzed in detail.

Applying the BeiDou Navigation Satellite System in the acquisition of the coseismic deformation and the high-frequency dynamic displacement of the 2021 MW 7.4 Maduo earthquake

SUMMARY This study acquires the coseismic deformation field and the high-frequency dynamic displacement of the MW 7.4 earthquake that occurred in Maduo, China, on 2021 May 22, based on the BeiDou Navigation Satellite System (BDS), and the comparison with the results obtained by the Global Positioning System (GPS) reveals that the two systems are certain differences in their ability to acquire the coseismic deformation field. The maximum difference in the horizontal coseismic deformation is <5 mm, and the maximum difference in the vertical coseismic deformation is 8.7 mm. The dynamic displacement waveforms of the 2021 MW 7.4 Maduo earthquake acquired by BDS and GPS are very similar, which confirms that BDS can acquire ground-shaking images with an accuracy comparable to that of GPS. Based on the empirical relationship equation of the peak ground displacement (PGD) and moment magnitude (MW), this study verifies and calculates both the MW of the 2021 MW 7.4 Maduo earthquake and the error and finds that the MW can be quickly and accurately obtained by using the empirical PGD and MW equations, and this MW value can be used as a supplementary means of calibrating the MW of the large earthquake early warning systems, which can be quickly determined by seismic wave data. Finally, by comparing the slip distributions inverted from the BDS and GPS coseismic deformation fields, this study finds that BDS is equally effective as GPS.

Displacement sensorless control of a bearingless induction motor based on modified high-frequency signal injection

ABSTRACT To enhance the rotor radial displacement identification accuracy of a bearingless induction motor (BIM) in displacement sensorless control, a displacement sensorless control strategy based on modified high-frequency signal injection is proposed. Firstly, a BIM flux linkage equation is analyzed, and by integrating the principle of the conventional high-frequency signal injection method, a self-detection model for the BIM rotor radial displacement is established. Secondly, a cascade notch filter and a low-pass filter are used in the signal demodulation process of the modified high-frequency signal injection. This strategy enables the DC component relevant to rotor displacement to be extracted from the high-frequency current，and obtains the rotor radial displacement equations. Finally, compared with the conventional high-frequency signal injection control, the proposed BIM displacement sensorless control system based on the modified high-frequency signal injection has higher displacement identification accuracy under different operation conditions, such as no-load, sudden load, and radial displacement mutation, as shown in simulation and experimental results. Overall, the motor performs well in terms of speed and suspension.

Modeling and Simulation of Elastohydrodynamic Lubrication in Spur Gears

An analysis using computational modeling by finite elements of the phenomenon of elastohydrodynamic lubrication (EHL) was carried out for a transmission system of pinion gear in a crankcase with partial filling lubrication. The analysis utilized tribological studies describing the contact behavior characteristics of solid surfaces with the lubrication film caused by dragging and splashing. Furthermore, the characteristics of the Reynolds-Hertz model for this type of phenomena are described, as well as the equations of elastic deformation and elastic displacements along with the geometry of the non-concordant bodies in contact. This was done by modeling the Lagrangian-Eulerian type for non-Newtonian fluid, implementing multiphysics coupling methods. The pressure profile of the lubricant films, the temperature reached by the lubricant, and the von Mises stress at the contact were obtained, showing a good approximation with the related results, indicating a range of 30 MPa to 900 MPa of pressure in the lubricant film and von Mises stress ranging from 30 MPa to 100 MPa in the contact area of the gear tooth.

Ultrasonic tool shank with multiple vibration mode for micro-nano drilling: Design, optimization and experiment

The machining effects of hole drilling is limited by merely changing the amplitude. Adjusting ultrasonic vibration frequency to expand the matching range of machining parameters optimization is necessary to improve hole drilling effects. Therefore, the design, optimization, and experiment evaluation for a novel integrated, fined, low-cost dual-frequency ultrasonic tool shank is proposed, which can work in three vibration modes for micro-nano precision drilling. The principle of the dual-frequency ultrasonic transducer is introduced. The electro-mechanical equivalent circuit model with step horn is applied to study the dual-frequency characteristics. The transducer displacement equation model based on the wave equation theory is deduced to find the shared vibration node of the first and third frequency. Sensitivity analysis by equivalent circuit model and Finite element method (FEM) simulation of the key parameter is both conducted to optimize ultrasonic transducer geometry dimension. The vibration node of mounting flange is specified in an identical position for the dual vibration modes. A prototype of ultrasonic tool shank is fabricated and evaluated in experiments. It demonstrates that the dual working frequency of the ultrasonic transducer is 34.9 kHz and 104.5 kHz in the first and third mode resonant vibration, which is benefit to the low and high frequency drilling. Three working modes of the low, high, and coupling frequency vibration is measured by a self-developed ultrasonic generator. When 40 V voltage is excited, the vibration amplitudes of the low and high frequency are 17.8 µm, and 9.3 µm, respectively, which can meet with the drilling requirement. Moreover, the coupling vibration with the low and high mode are excited simultaneously, and the motion trajectory of the coupling vibration is observed in a “M” shape, which is a novel trajectory for the precision micro-nano drilling. Micro drilling experiments show that the 35 kHz ultrasonic vibration drilling can achieve better machining effect than conventional drilling (CD). The 105 kHz ultrasonic drilling outperforms the CD, 35 kHz and coupling ultrasonic drilling at cutting force reduction. The coupling ultrasonic vibration drilling achieves the lowest chipping diameter. It is to be noted that the 105 kHz-8 µm produce the fined smoothly surface. The 35 kHz, 105 kHz and coupling ultrasonic drilling can produce better surface micro morphology with less quantity and size of craters than that of CD.

A simple recovery post-processing of stresses and displacements in Reissner–Mindlin analysis of anisotropic laminated plates

The bending problem of anisotropic laminated plates is considered, modeled with first order shear deformation (FSDT) kinematic model and approximations obtained from the Generalized Finite Element Method (GFEM/ XFEM). A procedure is developed to recover the transverse normal and shear stresses and all displacement components, with improved variations across the laminate thickness, generating a complete three-dimensional approximate solution of the problem. The procedure starts with the results issuing from direct computations of in-plane stresses and displacements obtained by the 2D kinematic and constitutive equations. The recovered fields are obtained to, approximately, enforce local equilibrium, constitutive and strain–displacement equations in their three-dimensional forms, and interlaminar continuity. The general procedure considers inertia forces and von Kármán non-linearity. Corrections are made to impose the necessary 3D boundary conditions in both faces of the laminate. The easy way the GFEM admits basis function enrichment, whether by singular, discontinuous or by higher order p-enrichment, on a fixed mesh, makes the entire recovery procedure straightforward and non-iterative. The recovered fields accuracy is demonstrated in standard problems against exact solutions from three-dimensional elasticity and FEM reference approximations. Up to the author’s knowledge, the presented strategy is novel in the published literature of non-iterative post-processing methods. It provides a simple mean to obtain all stress and displacement component approximations necessary to application in many complete 3D local failure theories.

In-plane and out-of-plane vibration analysis of laminated composite frames with warping effects

This paper addresses the free vibration analysis of a general planar frame structure, consisting of an assembly of laminated composite beam members. A mathematical model is established for the beams by introducing a novel displacement field, which encompasses the influence of shear deformation, rotary inertia, material coupling, warping phenomenon, as well as in-plane and out-of-plane deformations. By incorporating these various factors, a comprehensive representation of the beam's behavior is achieved. The continuity equations for displacements and rotations of two adjacent members are formulated to derive the equations for the entire frame structure. The assembled equations are solved using the Finite Element Method (FEM). The numerical results obtained are compared with existing results in the literature and new results from 3D models in ANSYS. The present results for torsional modes are closer to the ANSYS results than the results of previous studies. An extensive analysis is performed to investigate the impact of various parameters, including stacking sequence, frame angle, relative frame length, and anisotropy ratios on the system response.

Ion-Induced Swelling Behavior of Articular Cartilage due to Non-Newtonian Flow and Its Effects on Fluid Pressure and Solid Displacement.

In this study, we examine the behavior of articular cartilage equilibrated in a salt (NaCl) solution during non-Newtonian fluid flow that follows an Ostwald-de Waele model. A linearly elastic and isotropic rectangular strip of cartilage is considered for analysis. A continuum theory of mixtures has been employed to develop a coupled system of partial differential equations for the solid displacement and the fluid pressure by considering the important factor of the ion concentration by assuming the cartilage as a deformable porous media. The coupled system of partial differential equations is solved using the numerical method named method of lines. In most cases, shear-thinning fluid is compared to the shear-thickening fluid to magnify the difference. Graphical results show that shear-thickening fluids bring more solid deformation and shows less fluid pressure in comparison to the shear-thinning fluids.

Study on the imbibition production mechanism and main controlling factors of low-permeability reservoir.

With high water cuts and high permeability reservoirs entering the middle and late stage, many old oil fields in China are facing the dilemma of oil and gas resource exhaustion, while low-permeability reservoir resources remain very rich. As an oil recovery technology, imbibition displacement plays an important role in the exploitation of low-permeability reservoirs and prolongs the production cycle of old oil fields. Based on the imbibition kinetic equation, the mechanism and process of oil displacement in imbibition recovery are explained in this paper. Based on the characteristics of small particle size and high activity, the influence of temperature, viscosity, permeability, and salinity on the degree of imbibition recovery was studied. The results show that the imbibition production of low-permeability reservoirs mainly depends on capillary force, which firstly drains oil from small pores to large pores, and then drains it by buoyancy. Using nanofluids for imbibition oil recovery, a higher recovery effect of up to 45.59% can be achieved. The change in different external conditions will affect the imbibition recovery effect. The imbibition recovery efficiency increases with the increase in temperature, decreases with the increase in oil viscosity, and increases with the increase in core permeability. The increase in salinity has an inhibiting effect on the imbibition recovery efficiency. Based on the multivariate analysis of variance, the change in permeability has the greatest influence on the degree of imbibition recovery, and the change in viscosity has the least influence on the degree of imbibition recovery.

Adaptive meshfree method for fourth-order phase-field model of fracture using consistent integration schemes

An adaptive fourth-order phase-field model is proposed using consistent element-free Galerkin method (EFG). Convenient construction of high-order interpolation functions in EFG method is fully utilized. Requirement of C1 continuity of shape functions is satisfied in fourth-order phase-field model. Two different consistent integration schemes with high efficiency and high accuracy, namely quadratically consistent one-point integration scheme and three-point integration scheme, are employed to evaluate the fourth-order phase-field equation and displacement equation, respectively. To reduce computational cost, the adaptive strategy of local background mesh refinement is established on basis of strain energy history and phase field. Background mesh is refined via the insertion of nodes at the midpoints of each side of the integration element. Comparing with EFG method, computational accuracy and efficiency are enhanced by the proposed method. Load-displacement response, the smoothness of the resulting stress and crack paths are predicted well.

Modelling and Investigation of the Dynamic Behavior of a Penny-Shaped Interface Crack in Piezoelectric Bimaterials

In this section, the dynamic propagation behavior of a penny-shaped interface crack in piezoelectric bimaterials is analyzed. The objective of this paper is to use the boundary conditions of the penny-shaped interface crack to study the dynamic propagation of the crack under the action of load, so as to provide some valuable implications for the fracture mechanics of the piezoelectric bimaterials and simulate the interface crack between piezoelectric bimaterials, it is necessary to establish a suitable model and give appropriate boundary conditions according to the actual situation. The elastic displacement and potential equations are constructed according to the structural characteristics of the circular crack. In the case of a given displacement or stress, the Laplace transform and Hankel transform are used to simplify the problem into an integral equation with unknown functions. According to the boundary conditions, the corresponding unknowns are obtained, and the closed solution is derived. The results show that the fracture toughness of a penny-shaped interface crack in piezoelectric bimaterials is related to the thickness of the material, the impact time, the material characteristics, and the electric field. At the same time, it can be found that different materials have different roles in the crack propagation, so it is very important to study the crack opening displacement (COD) intensity factor of the crack for safety design.

On a Cahn–Hilliard system with source term and thermal memory

A nonisothermal phase field system of Cahn–Hilliard type is introduced and analyzed mathematically. The system constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. It couples a Cahn–Hilliard type equation with source term for the order parameter with the universal balance law of internal energy. In place of the standard Fourier form, the constitutive law of the heat flux is assumed in the form given by the theory developed by Green and Naghdi, which accounts for a possible thermal memory of the evolution. This has the consequence that the balance law of internal energy becomes a second-order in time equation for the thermal displacement or freezing index, that is, a primitive with respect to time of the temperature. Another particular feature of our system is the presence of the source term in the equation for the order parameter, which entails additional mathematical difficulties because the mass conservation of the order parameter is lost. We provide several mathematical results under general assumptions on the source term and the double-well nonlinearity governing the evolution: existence and continuous dependence results are shown for weak and strong solutions to the corresponding initial-boundary value problem.

Hydroelastic Response of a Moored Floating Flexible Circular Structure Applying BIEM

A hydroelastic model associated with the interaction between a surface wave and a floating circular structure connected with mooring lines in finite water depth is developed using BIEM. The BIEM solution is achieved using free surface Green’s function and Green’s theorem. Furthermore, the algebraic equations for circular structural displacement are derived from the integro-differential equation. The correctness of the BIEM code is verified with the results of shear force and deflection amplitude existing in the literature, and the hydroelastic response of the circular structure is analyzed. The comparison results show a good level of agreement between the present results and those from other calculations. It is observed that the shear force, bending moment, and deflection decrease for higher values of stiffness of the mooring lines. The current study may be supportive of the visualization of the effect of mooring stiffness and to generalize articulated circular structure models for ocean space utilization.

Numerical and Experimental Study into Paper Compression Test.

The study aims to present the results of paper compression under an axial load. Different heights of samples subjected to compression were taken into account. The main goal of the analysis was to determine experimentally the maximum compression load. In addition, numerical models based on the finite element method (FEM) were validated to refer to empirical results. The performed numerical simulations were founded on Green-Lagrangian nonlinear equations for large displacements and strains. The progressive failure of the compressed orthotropic material after exceeding maximum stresses was based on Hill's anisotropy theory. Nonlinear calculations were conducted by using a typical Newton-Raphson algorithm for achieving a sequence convergence. The accuracy of the developed model was confirmed experimentally in compression tests. The technique of analysing the shape of the compressed paper sample on the basis of images recorded during the measurement was used. The obtained test results are directly applicable in practice, especially in the calculation of the mechanical properties of corrugated cardboard and in determining the load capacity of cardboard packaging. Knowing the maximum compressive stress that packaging paper can withstand allows packaging to be properly designed and its strength assessed in the context of the transport and storage of goods.

On the Flow Physics During the Transition of a Variable Pitch Fan From Nominal Operation to Reverse Thrust Mode

Abstract The flow field during the transition of variable pitch fans (VPFs) from nominal operation to reverse thrust mode at typical “Approach Idle” engine power setting and aircraft touchdown speed of 140 knots is described in this work. An integrated airframe-engine-VPF research model that features a future 40,000 lbf geared high bypass ratio engine installed on a twin-engine airframe in landing configuration is used to explore the flow field in a fully transient unsteady Reynolds-averaged Navier–Stokes (URANS) simulation with imposed wall motion. A novel methodology that implements an adaptation of a mesh displacement equation to mimic the fan blade airfoil rotation is developed. The implementation of this method with gradual, small-step deformation along with an automated mesh update routine enables a high quality, near “real-time” simulation of the complete transition. The flow field during transition is characterized by the evolution from full typical forward flow through the engine to the development of massive recirculation regions at the feather pitch setting and finally to development of a reverse flow from the bypass nozzle to the fan passages. In the paper, the transient development of the various flow features through different stations of the engine flow path apropos the fan blade airfoil rotation to reverse thrust mode are discussed in detail. Also, the temporal development of the mass flow ingested through the engine, airframe decelerating force, and distorted flow at the core engine inlet are described. A hitherto unresolved fan power peaking during the middle of the transition and higher power requirement at reverse thrust mode is captured. The effect of fan rotational speed in terms of engine power setting and the aircraft touchdown velocity on the transition flow physics is explored. A comparison of this fully transient approach with discrete steady-state runs for different stagger angle settings is presented. The flow physics during transition to reverse thrust mode as described in this study is critical in understanding the feasibility of using VPF for reverse thrust in future aircraft. The new capability to study the transition in a fully transient simulation can be used as a design development aid to define design and control characteristics of the reverse thrust VPF.

Self-weight and thermal bifurcation analysis of functionally graded vertical annular plates

The present paper thoroughly investigates the influence of structural weight on the thermal buckling of standing circular and annular plates. The structure is composed of symmetrically functionally graded materials (FGM) in terms of its thickness. The first-order theory of shear deformation along with nonlinear strain–displacement equations has been employed to derive the governing equations of the FG plate. First, the asymmetric pre-buckling path of the structure under the influence of weight and heat is calculated by a semi-analytical method. The plate's stability relations are extracted using the adjacent equilibrium criterion and virtual work principle. Then, the stability equations are analyzed by the 2D generalized differential quadrature (GDQ) method. Parametric results are presented based on the analysis of the FGM power-law index, geometric ratios, and edge constraints on the buckling of the vertical plate. The study reveals the critical power-law index and thickness at which the structure becomes unstable, shedding light on the previously neglected calculation of weight-induced buckling in vertical structures. Also, the effect of weight on the buckling mode shapes of heavy plates and disturbing its symmetry is demonstrated. Additionally, this study presents the first investigation into the effect of weight on the thermal stability of annular FG plates.

Melon Robotic Grafting: A Study on the Precision Cutting Mechanism and Experimental Validation

The grafting machine cutting step is the core step of the grafting process. The existing grafting machine cutting mechanism adopts fixed angle cutting and manually adjusts the cutting angle based on experience, and the cutting angle is not definite for rootstock cutting in different growth periods. In this paper, we proposed a method to guide the precise cutting mechanism according to internal and external parameters of melon rootstock at a specific period. First, we constructed a cutting model based on internal and external characteristics of rootstock cutting in the growth period of “two leaves and one core” and clarified the safe cutting area. Second, we designed a rotary precision cutting mechanism for rootstock cutting with automatic angle adjustment and constructed the displacement equation of the cutting trajectory of the cutter according to the cutting model. Last, we examined the cutting effect of the precision cutting mechanism and determined the precise cutting angle of the rootstock cutting mechanism in the growth period. Finally, the cutting effect of the precision cutting mechanism was examined, and the precision cutting angle of the rootstock during the growth period was determined. A comparison test between the precision cutting mechanism and the traditional cutting mechanism was carried out, and visual images of the incision were captured and analyzed. The results show that under the five horizontal cutting angles in the safe cutting area of rootstock, the length of the cut surface is inversely proportional to the cutting angle, and the flatness of the cut surface is directly proportional to the cutting angle. Comprehensive evaluation of the length of the cut surface, the cutting success rate, and the quality of cutting revealed that the average cutting angle of the precision cutting mechanism in the safe cutting area (26°) is better than that of the traditional cutting mechanism. It meets the technical requirements of the cutting technology for mechanically grafted rootstocks. The results provide a reference for studying new rootstock precision cutting mechanisms and cutting angle adaptive control models.