Force Equilibrium Equations Research Articles

Experimental studies on lateral bearing capacity of modified tripod suction caissons for offshore wind turbines in silty clay

In recent years regular tripod suction caissons are commonly used in offshore wind farm projects because of economic feasibility and environment-friendly work principles. The design of a modified tripod suction caisson composed of three caissons with different diameters can improve the capacity behaviour in the direction of strong wind within a considered construction cost. This paper carried out model tests to study the lateral bearing capacity of modified tripod suction caissons in silty clay. Based on the equilibrium equation of vertical force of modified tripod suction caissons the average vertical subgrade reaction modulus is proposed to investigate the relationship between it and rotation, which is of paramount importance to the nonlinear lateral bearing capacity. The results show that the range of bearing capacity factors increases with the increasing effective diameter-height ratio indicating that modified tripod suction caisson with reasonable combinations can effectively improve its bearing capacity in a certain loading direction. The sum of the bearing capacity factors of three monopod suction caissons falls in the range of the capacity factors of the tripod foundation consisting of the above caissons. Suction under the lid increases with the increasing lateral load and seems to be not related to the combinations of suction caissons with different diameters. By fitting the experimental data, it is found that there is a power exponential relationship between the average vertical subgrade reaction modulus and rotation of the foundation.

Numerical simulation and model development of drag coefficient of bubbles in gas-liquid metal two-phase flow

The occurrence of a steam generator tube rupture (SGTR) accident in a lead-bismuth cooled fast reactor results in the formation of steam bubbles in the liquid lead-bismuth eutectic (LBE). This may degrade heat transfer and power transients in the reactor core due to the migration and accumulation of steam bubbles. To investigate the dynamics of steam bubbles flowing in liquid LBE, it is essential to develop an accurate model for the bubble drag coefficient. In this paper, a three-dimensional numerical model is first established to simulate the injection of high-pressure steam bubbles into a high-temperature LBE molten pool. The model is based on the CLSVOF method. By analyzing the trajectory, velocity, and diameter of bubbles, and combining them with the force equilibrium equation for bubbles, the values of the drag coefficient for bubbles are determined. On this basis, the suitability of current empirical drag models for bubble migration in LBE is evaluated. Finally, the optimal drag coefficient model is selected and further improved. Results reveal that the prediction error of the optimized model for the bubble drag coefficient in liquid LBE is within ±15 %.

Research on Load Estimation for Commercial Trucks Based on Air Suspension State Parameters

For commercial trucks, accurate vehicle load information is crucial to preventing overloading and uneven load distribution, thereby improving transportation efficiency and reducing transportation costs. Existing research on vehicle load estimation primarily focuses on tires, axles, frames, and leaf springs, with limited studies on load estimation for vehicles equipped with air suspension. This paper presents the design of a vehicle onboard weighing system for civil commercial trucks based on the state parameters of air springs and proposes a matching method for vehicle load estimation. The system utilizes laser distance- and gas pressure sensors to collect the state parameters of the air springs. It estimates the vehicle load and displays the load information in real time on the onboard display. The vehicle load estimation method was explored through prototype testing and theoretical derivation. A fitting model was constructed using surface fitting techniques to establish the relationship between the effective contact area and the height variation and capsule pressure of the air spring. Subsequently, a load prediction model was built based on the force equilibrium equation of the air springs. Finally, the construction and accuracy of the load prediction model for a specific type of diaphragm air spring were verified using finite element analysis. The results demonstrated that the average relative error of the load prediction model could be controlled within 1%, meeting the high accuracy requirements of the vehicle weighing system.

Sub-global equilibrium method for identification of elastic parameters based on digital image correlation results

In this work, a new, simple method is presented, which enables identification of material properties of solids basing on the digital image correlation (DIC) measurements. It may be considered as a simplified alternative of low computational complexity for the well-known finite element model updating (FEMU) method and virtual fields method (VFM). The idea of the introduced sub-global equilibrium (SGE) method is to utilize the fundamental concept and definition of internal forces and its equilibrium with appropriate set of external forces. This makes the method universal for the use in the description of a great variety of continua. The objective function is the measure of imbalance, namely the sum of squares of residua of equilibrium equations of external forces and internal forces determined for finite-sized part of the sample. It is then minimized with the use of the Nelder–Mead downhill simplex algorithm. The efficiency of the proposed SGE method is shown for two types of materials: 310 S austenitic steel and carbon-fiber-reinforced polymer (CFRP). The proposed method was also verified based on FE analysis showing error estimation.

Continuum strain of point defects

We propose a new Regularized Green's Function Method (RGFM) derived from electron densities and captures the disturbance due to point defects, successfully extending the elastic strain determination to the lattice scales. The RGFM circumvents the use of concentrated point forces, which results in unrealistic singular fields. The objective is to determine the force variations at the atomic scales from Ab-initio calculations as Quantum Mechanical Force Density (QMFD). QMFD encodes the information regarding the electronic structure of the defect since it is related to the gradients of electron wavefunctions generated from the Density Functional Theory (DFT) calculations. Once the QMFD is calculated, RGFM solves the force equilibrium equation via Fourier transforms to compute elastic fields. Therefore, the present treatment reflects the electronic structure of the atomic positions, hence the complex elastic deformations and interactions at short range, which represents a significant advancement compared to previous studies. The RGFM can also capture the long-range fields since it calculates the decay of the force fields away from the nuclei. Previous theories, such as the elastic dipole method, face two main shortcomings: they can only handle the long-range elastic fields, and contain an unphysical singularity at the center of the point defect. Our novel derivation of distributed forces addresses both challenges, i.e., it renders non-singular displacement and strain fields at the nuclei, and can also describe the elastic response accurately far from the nuclei center. The application of the method to the NV (nitrogen-vacancy) center in SiC is demonstrated in comparison with the elastic dipole theory, showing the advantages of the present methodology.

Developing Dynamic System Responses Model Based on Nanotechnology by Incorporating the Finite Element Analysis

The dynamic model of complex structure has set incredible progression with the use of finite element analysis (FEA), especially in the field of vibration control and nanotechnology. Main focus of this research is to conduct a frequency analysis in ANSYS and a state space model of the beam under the platform of MATLAB. This allowed a system to develop an active dynamic model for vibration control. To enhance test and analysis of the cantilever beam of Aluminium 6061T6, a new method has been proposed to predict the dynamic response of a cantilever beam under sinusoidal base excitation. This was achieved by creating an analytical model for the cantilever beam using moment and force equilibrium equations. The authenticity of the proposed method was made by comparing the results with experimental data. Additionally, to control vibration, a Proportional-Integral-Derivative (PID) controller was developed using the system model. The FEA in the ANSYS platform provided a cantilever beam model. Also, its mathematical modelling has been done. The proposed method utilizes a novel disturbance rejection control scheme that eliminates an unknown disturbance. Experimental results indicate that the control system results in gradually decreased beam vibration amplitude. The state space approach discussed in the work could be a valuable tool for studying the behaviours of nanomaterials.

Analytical model for prestressed glulam beams reinforced with unbonded steel bars

Glulam beams reinforced with bonded steel bars are a common form. The study describes theoretical models of unbonded reinforced and prestressed reinforced glulam beams for assessing the reinforcement ratio and prestress level range, and for predicting the effect of unbonded steel bars on the flexural capacity of the beam. Considering the stress changes of beam cracking and cross-sectional stress, the minimum and maximum reinforcement ratios were determined. On this basis, the influence of the service load on the beam in the actual engineering was explored, and the force equilibrium equation was established to derive a reasonable prestress range related to the reinforcement ratio. Combined with the glulam constitutive model, and analyzed the strain distribution of unbonded steel bars was to establish the flexural capacity prediction model. The accuracy of the model was verified by the relative error between the prediction and test values ranging from-11.78% to 8.84%.

Dynamics of acoustically-induced droplet instability

The advancement of the theory of droplet stability in the acoustic field is of significant value in improving ultrasonic atomization and ultrasonic levitation technology. In this work, in order to reveal the detailed mechanism of acoustic droplet instability and give the instability criterion for easy application, the dynamics of droplet instability in standing wave acoustic field (19.8 kHz) is studied by combining practical experiment, theoretical derivation and numerical calculation. The acoustic instability of the droplet occurring near the wave nodes is mainly manifested in two typical modes: disk instability and edge-sharpening instability. The appearance of these two instability modes depends on the relative magnitude of the standing wave field strength. Specifically, with the gradual enhancement of the intensity of the standing wave field, the instability mode of the droplet will gradually change from disc instability to edge-sharpened instability.The droplets show obvious self-accelerating expansion in the equatorial plane in the instability process. The positive feedback between the droplet aspect ratio and the negative pressure of acoustic radiation at the equator of the droplet is the reason for the above self-accelerating behavior. The theoretical results obtained through deduction indicate that the amplitude of the negative acoustic radiation pressure at the droplet equator is proportional to the square of the droplet aspect ratio. The surface tension of the droplet is the main factor hindering its deformation, while the acoustic radiation suction at the equator is the main factor driving the deformation of the droplet. Based on this, the force equilibrium equation of the droplet interface is established, and the dimensionless criterion of acoustic droplet instability, i.e. the acoustic Weber number <i>We</i><sub>a</sub>, is derived. When <i>We</i><sub>a</sub>≤1, the droplet interface stays in equilibrium, and when <i>We</i><sub>a</sub> >1, the equatorial acoustic suction is larger than the surface tension, and the droplet instability occurs, and the average error between the experimental results and the theoretical results is only 9%.

Design of a deployable aerodynamic decelerator based on a tensegrity structure

Deployable aerodynamic decelerators are more suitable for planetary exploration missions than rigid aerodynamic decelerators. Their sizes are less limited by the launch vehicle shroud, allowing heavier payloads to be delivered to the surfaces of planets. Aiming at the problem of lightweight designs in deployable aerodynamic decelerators, a deployable tensegrity structure composed of two frustum tensegrity units and a prism tensegrity unit is proposed. To design the deployable tensegrity structure, a self-equilibrium model that is especially suitable for the structure with rotational symmetry is established according to prestressability conditions and force equilibrium equations. Based on the deployable tensegrity structure, the deployment mechanism, with the action of drive units, is designed to realize the transformation from the stowed state to the deployed state of the decelerator. The tensegrity structure always has rotational symmetry to make the deployment process highly synchronized due to the fact that drive units can work simultaneously. During the deployment process, the transverse distance and longitudinal distance are used to measure the deployment degree of the tensegrity structure. The types of axial forces for members are used to determine whether the tensegrity structure is stable. The result shows that the deployment process has continuity and that the tensegrity structure always stays stable.

A novel separation density model based on force analysis of coal-based oil shale in pulsed fluidized beds

ABSTRACT It is important to study the theory and technology of coal-based oil shale dry beneficiation in China. Traditional separation density models are obtained through particle force analysis, but most studies tend to ignore the influence of medium resistance and bubble disturbance, which seriously limit the predicted accuracy of the separation density model. Therefore, the force on simulated mineral particles (steel balls) in pulsed fluidized bed (PFB) under different operating conditions was tested. It was found that the force decreases with the increase of gas velocity and distance from the distributor. The change of pulsation frequency has less influence on force. The force equilibrium equation was established and verified to be within 10% error; Finally, we obtain a new separation density model of PFB. This is of great importance for the industrialization and scale-up of dry beneficiation.

Analysis of mechanical characteristics and normal operating conditions of magnetic seals

Aiming at the leakage failure problem caused by the accidental disengagement of the sealing interface due to the failure of the magnetic seal force balance system, taking a certain type of aero-engine magnetic seal as an example, the mechanical analysis model of the magnetic seal is established, and the mechanical conditions for normal sealing work are deduced according to the set of equilibrium equations of force. The analysis process of the normal operation is proposed, and the calculation method of the key mechanical parameters of the magnetic seal is determined. On this basis, the influence of operating conditions on the normal operating mechanical conditions is analyzed. The results show that the primary sealing face contact pressure, the maximum contact pressure of the secondary sealing face, and the traction torque of the rotor determine the normal operating condition of the magnetic seal. Excessive chamber pressure and rotational speed will increase the contact pressure of the primary sealing face, leading to increased wear of the face and slippage between the O-Ring 1 and the shaft; too little differential pressure and rotational speed will lead to the disengagement of the graphite ring/stator, which ultimately leads to the failure of the seal. Severe vibration reduces the contact pressure of the primary sealing face and increases the risk of the primary sealing face disengagement.

A novel framework for ascertaining the safety factor and critical slip surface of 3D slopes reinforced with piles

The pile is a widely used reinforcement device in geotechnical engineering. For the present paper, we propose a new solution for calculating the safety factor (SF) of the 3D slope reinforced with piles and determining the critical slip surface (CSS) in the framework of minimum potential energy method. In particular, the mobilized shear stress on failure surface is determined by the equilibrium equation of forces operating on slip body. Meanwhile, the moving direction of 3D slope is equivalent to the direction of combined force acting on slip body. Additionally, the present method is combined with the genetic algorithm to identify CSS, and a novel criterion is proposed to locate the shape and position of CSS of a 3D slope using the total potential energy of landslide system as the objective function. The validation and comparison of results indicate that the SF obtained by the introduced method are close to reference solutions, and SF shows an obvious negative correlation with sliding direction (SD). It is also found that the support effect of pile is related to the pile location and the pile density. Meanwhile, the optimal pile position depends on the shape of failure surface and shear strength parameter of soil. Moreover, the effects of some parameters on SD of 3D slopes reinforced with piles are also studied.

Improving shape formation under conditions of plane tensile stress

Thin-walled axisymmetric truncated parts made of sheet billets are actively used in rocket and aerospace engineering. Improvement to their shape formation, based on directed material thickness change will ensure the production of parts with minimum thickness variation. This will also enable aviation and space industry enterprises to attain leading positions, as well as reduce labor costs. This work studies the possibility of obtaining thin-walled axisymmetric parts of truncated tapered shape using one of the methods of sheet metal stamping under flat tensile stress conditions (flanging). The mechanism was identified and the analysis of the stress-strain state of the billet during deformation was carried out. This takes into account the minimizing of the difference between the specified and technologically possible thicknesses. A mathematical model was developed to consider the shaping method based on the process of flanging. Theoretical studies were based on the principles of the plastic deformation theory of sheet materials. This was achieved by the following factors: approximate differential equations of force equilibrium; equations of constraint; plasticity conditions; and fundamental constitutive relations under given initial and boundary conditions. The process of flanging was simulated using the LS-DYNA software package with the following initial data of a conical billet made of 12Kh18N10T steel: cone angle 16.4°, thickness Sbillet = 0.3 mm. The aim was to eliminate errors in designing a tool for future implementation of the method on a manufactured die tooling, as well as to confirm the theoretical conclusions on the selection of technological parameters and achieve minimal thickness variation. The steps of computer modeling are presented, indicating the main process parameters such as material model, mechanical characteristics of the workpiece material, type of elements, kinematic loads, conditions of contact interaction of elements with each other, etc.

Solving the general 3-D safety factor by combining Sarma's idea with the assumption of normal stress distribution over the slip surface.

This study proposes a method for determining 3-D limit equilibrium solutions. The method, inspired by Sarma, introduces the horizontal seismic coefficient as a slope failure parameter and implements a modification of the normal stress over the slip surface. Four equilibrium equations are used to solve the problem without compromising the accuracy of the calculations: three force equilibrium equations in the x, y, and z directions and a moment equilibrium equation in the vertical (z) direction. The reliable factor of safety can be determined by calculating the minimum value of the horizontal seismic coefficient. Furthermore, we analyzed several typical examples of symmetric and asymmetric slopes, finding good consistency with the existing literature. This consistency indicates the reliability of the factor of safety we obtained. The proposed method is favored due to its straightforward principle, convenient operation, fast convergence, and ease of programming.

A coupled SBFETI-BDEs method for solving 2D static and dynamic contact problem with friction

In this study, we propose a coupled approach called the SBFETI-BDEs method for solving the 2D elastic frictional contact problem. The contact system is decomposed into multiple subdomains, similar to the finite-element tearing and interconnecting (FETI) algorithm. Each subdomain can be regarded as a scaled boundary finite element (SBFE), where only the boundaries are discretized. To avoid computing the global contact flexibility matrix using the force method, we consider the generalized inverse of the stiffness matrix for each subdomain as a subcontact-flexibility matrix. We uniformly formulate the contact equations, displacement continuity equations, and the equilibrium equations of force for the floating subdomains, as a B-differentiable equations (BDEs) system, which we solve using the B-differentiable damped Newton-Raphson method that possesses a global convergence property. The proposed method offers increased computational efficiency without compromising accuracy when compared to the original BDEs algorithm without domain decomposition. We present the dynamic scheme of the SBFETI-BDEs method within the framework of the HHT time-integration scheme, where the equilibrium equations of force for the floating subdomains disappear due to their nonsingular equivalent stiffness matrices. Numerical examples are provided to demonstrate the convergence, accuracy, efficiency, and capability in solving engineering problems of the proposed method.

Research on force model and characteristics of large wind turbine pitch system based on SCADA data

In large-scale wind turbines, the force state of the pitch system greatly influences safe operation and service life. This paper provides a novel method to estimate blade pitch load, bearing friction torque, and motor pitch torque. In this method, the force equilibrium equations are established by investigating the force of the pitch system under multiple operating conditions. The multidimensional BIN method is employed to classify the supervisory control and data acquisition (SCADA) data of wind turbines into several intervals. The multidimensional scatter data is processed in a single-valued way. Then, the estimating model of the pitch system forces is established by combining the obtained data and the equilibrium equations. Taking a 2 MW wind turbine as an example, the variation characteristics of blade pitch load, bearing friction torque, and motor pitch torque under multiple operating conditions are analyzed. Some interesting and valuable conclusions are obtained. For example, when the wind speed increases, the blade pitch load increases significantly in the maximum wind energy tracking region, but there is no obvious change in the observed constant power output region. The wind speed and azimuth have little effect on the bearing friction torque. The variation trend of motor pitch torque is consistent with that of blade pitch load in the maximum wind energy tracking region.

A hybrid transient/quasi-static model for wet clutch engagement

This work presents a comprehensive formulation of the equations governing the lubrication and the lock-up phases of an engaging wet clutch. A finite element model is established through the coupling of the Reynolds equation of the fluid film model with the dynamic force and torque equilibrium equations of the two-disc system. To solve the problem, an efficient hybrid simulation technique is introduced, capable of capturing the rapidly evolving transient phenomena at the start of the engagement phase, which later settles into a quasi-static one as the film thickness asymptotically decreases. The simulation tool is benchmarked against previously published results and is later used to investigate the effect of a newly introduced groove configuration utilising variable-width grooves. The proposed geometry is compared against the well-known radial groove configuration on the engagement characteristics, resulting into decreased hydrodynamic peak force by approximately 20%. The supremacy of the proposed groove design over the standard radial geometry is further verified through a newly introduced groove design criterion that takes into consideration the relative effect of squeeze and torsional Couette flows during the hydrodynamic phase of the engagement. Finally, a comprehensive parametric study is conducted to capture the effect of the groove width and depth on the engagement variables.

Fracture initiation sequence of double-layer beam layers under three-point loading

The article deals with brittle failure of a two-layer beam under three-point loading conditions depending on the parameters under consideration — different proportions of thicknesses, Young’s moduli and strengths of both layers. Based on the equations of force and moment equilibrium, the dependences of the position of the beam neutral axis and its curvature are derived and the regions of parameters are determined, at which the failure begins earlier in the layer to which the load is applied than in the outer opposite layer to the load.

Exact solutions for Janssen effect of lattice disk packings

The rhombic lattice packing model proposed here clearly shows the micro causes of the Janssen effect and is solved analytically. Based on the analysis of force equilibrium equations with an exterior product method, a recurrence relation with a force transmission matrix Q is obtained which shows that a contact force will be transmitted and accumulated along the lattice vector direction and then be reflected by lateral walls, and eventually, reach a saturation value if the packing is frictional. A method based on tensor algebra and difference equation is proposed for the diagonalization of Q. Using roundup and remainder functions, we give the exact and approximate expressions which can describe the piecewise smoothness of the pressure function data given by numerical calculations. Numerical results also show the non-uniformity of pressure distribution. These quantitative results have certain deviations with Janssen model except for very small top load or large force reflectivity.

Application of controlled blasting demolition technology in ultra-high coaxial thin-walled steel inner cylinder reinforced concrete chimney

To achieve the goal of 'carbon neutrality', it is of great significance to accelerate the process of dismantling high-carbon and high-pollution thin-walled cylinder structures such as chimneys and so on. A 183 m ultra-high coaxial thin-walled steel-inner-cylinder reinforced concrete chimney (UCTS-RCC) was successfully demolished for the first time. To successfully dismantle the UCTS-RCC by controlled blasting demolition technology, the necessary structural mechanics theoretical analysis and the critical blasting designs rationality verification were carried out. Based on the equilibrium equation of force and bending moment, a theoretical model of incision angle selection was established to determine the incision design of the steel inner cylinder (SIC). The stability of SIC under the designed incision parameters was verified based on FEM statics analysis. The cutting experimental on the target steel plate was completed with a linear shaped cutter to obtain the cutting effect. The cutting process and the overall collapse process of the UCTS-RCC were simulated by LS-DYNA. Based on the fixed axis rotation differential equation, the synchronous toppling theoretical model were established to study the motion of the UCTS-RCC. The results showed that the maximum stress at the position of the directional window and the stresses of the remaining supports were less than the yield limit of Q235B steel after that several pretreatment windows were carried out, the SIC could maintain the stability of remaining structures. The cutting experimental results of the target steel plate using linear shaped cutter were basically consistent with the cutting simulation and the cutting effect satisfied the design requirements of instantaneous cutting separation of the supports. According to the obvious difference of material properties between Q235B steel and reinforced concrete, the ignition delay time of the SIC was set 2 s earlier than that of the RCC. According to the synchronous collapse simulations, it could be known that the ignition delay time setting was reasonable and the collapse motion state of the UCTS-RCC was approximately fixed axis rotation, which was in line with the established synchronous toppling theory model. The SIC and RCC finally successfully collapsed synchronously. The blasting scheme and the rationality verification methods were applied successfully in this project, which was of great significance to provide a reliable reference for such ultra-high thin-walled cylinder complex structural chimney.