Dynamic Equilibrium Equations Research Articles

Design and research of face gear central limited slip differential and analysis of its influence on vehicle performance

In this paper, a new design scheme of face gear central limited-slip differential (F-LSD) is proposed, which aims to unveil the technical veil of central limited-slip differential and fill the research blank of face gear application in the field of central limited-slip differential. Through the analysis of structural principle and motion characteristics, the dynamic equilibrium equation and locking coefficient model of F-LSD are established. The Simulink simulation model based on F-LSD dynamics and the torque distribution model of the whole transmission system are established. A 7-DOF vehicle dynamics model was established, and the performance of F-LSD vehicle in four conditions of acceleration, climbing, chassis twist sine wave and double line change was studied by using simulation software. At the same time, four groups of four-wheel drive vehicles with different central differential speeds were set as the control test group. The simulation results show that F-LSD system can effectively improve the vehicle's dynamic performance and extrication performance, and will not reduce the vehicle's handling stability, which provides a reference for the practical application of the face gear central limited slip differential.

Elastodynamic behaviors of steady moving straight dislocation within thin nano film

The elastodynamic dislocation behaviors are of great interest for understanding the performances of structural alloys under intense dynamic loading conditions. The formation, propagations, and interactions of dislocations (such as injected dislocation, accelerating dislocation, steady moving dislocation at high constant speed) are quite different from static dislocations. For steady-moving dislocation within the isotropic infinite medium, the effects of surface and interface on steady-moving dislocations within limited space are still known. In this paper, we investigate the elastodynamic image stress simulation of steady moving dislocation within film of limited thickness at constant speed using Eigenstrain theory, Lorentz transformation, and steady dynamic equilibrium equations. We propose an efficient solution method that involves complex Fourier series, transforming partial differential equations into ordinary differential equations, and ultimately into a set of algebraic equations in spectral space. The effects of dislocation speed and position near the free surface on the image stress of steady-moving climbing and gliding dislocations within the thin film are examined. The results show that relativistic effects are significant for certain dislocation configurations and stress components, whereas other stress components are less sensitive to relativistic effects near the transonic speed region.

Analysis on transient wave propagation in the soft matter of hydrogels

The precise control and structural design of the soft matter of the hydrogel subjected to dynamic loading require an in-depth understanding of its transient wave characteristics. However, for the complex solid–liquid coupling effects of the soft matter of hydrogels, the traditional single-phase wave model is no longer applicable. In this study, a transient wave propagation model that considers the solid–liquid coupling effects and its computational method is proposed. The wave-governing equations of hydrogels are derived based on the mass conservation and dynamic equilibrium equations, where the motions of solid and liquid phases are independently described. After transforming the governing equations into the equivalent weak form, numerical solutions for transient wave propagation in one- and two-dimensional hydrogels are calculated. The accuracy of the proposed model is verified by a comparison between semi-analytical and commercial finite element method solutions. We observe that the travelling and reflection processes of the two types of compression waves, P1 and P2, with different wave speeds are captured when the dynamic coefficient of permeability kf increases. Furthermore, the influence of solid–liquid coupling effects on the transient responses of hydrogels is discussed. The results show that the hydrogels exhibit the dynamic characteristics of a single-phase medium to a certain extent when kf is sufficiently small.

Seismic response of column‐supported silos considering granular–structure interaction

AbstractTo predict the seismic response of column‐supported silos (CSSs), the granular–structure interaction (GSI) analysis method is proposed with considering the combined effect of the friction between the particles–particles and the particles–silo wall. Using free‐body dynamic equilibrium equations, we reconstruct the mutual interactions between different grain portions and between the grains and the silo wall to develop the ideal calculation model of the CSS structure. Based on the analysis model, additional dynamic overpressure and the effective mass caused by the stored content interacting with the silo wall is obtained with different slenderness ratios and peak accelerations. The additional bending moment caused by the friction between the particles and silo wall is further quantified. To verify the reliability of the proposed method, we discuss some applicative examples by comparing the GSI method with other theories, Eurocode 8, and experimental results. Moreover, the along‐the‐height acceleration profiles of the silo wall and the ensiled content are analyzed according to the shaking‐table tests. The results show that the GSI method can match Janssen's theory well in the case of static pressure at slenderness ratios exceeding 1.0. The overpressure profiles along the height of the silo wall follow a nonlinear distribution, different from Eurocode 8. The bending moment obtained by predictive formulas agrees well with the experimental results for the CSS, indicating that the GSI method is reasonable. Some design and construction recommendations, including the maximum overpressure position, the reference range of the dynamic overpressure coefficient, and the reduction factors of the ensiled content mass, are proposed to facilitate the engineering applications of CSSs, considering different slenderness ratios.

Vibration-based methods for local damage identification of breathing cracks in truss-like structures

A vibration-based local damage identification method is proposed for truss-like structures with breathing cracks. Firstly, a bilinear model is established to represent the characteristics of breathing cracks, and the features of such cracks are detected through reciprocity and homogeneity tests on structural input-output signals. Since the crack is influenced by tensional and compressive forces on truss members, only one damage factor is required to quantify the stiffness reduction of each member. Subsequently, a dynamic equilibrium equation concerning the damage factor is constructed to identify the new load term. The integral equation method, which relies on the virtual work principle, is employed to obtain the displacement responses and loads under all tensile conditions. Finally, the mean value of cumulative damage coefficients for each truss member can be derived as their corresponding damage factor. Numerical examples involving a plane truss structure and a truss offshore platform are utilised to illustrate the effectiveness of this damage identification method. The results demonstrate exceptional precision and efficiency in accurately identifying various damages in truss structures under different excitations and signal-to-noise ratios. A actual space truss structure confirm that this damage identification method exhibits high-level precision in identifying single-member damages as well as multi-member damages with breathing cracks locally.

Free vibration analysis of laminated anisotropic doubly-curved shell structures reinforced with three-phase polymer/CNT/fiber material

The present work investigates the vibrational response of laminated anisotropic doubly-curved shell structures reinforced with Carbon Nanotubes (CNTs) short fibers. The fundamental equations are derived from a curvilinear reference system of principal coordinates, employing the Equivalent Single Layer (ESL) methodology. Furthermore, a general variation in laminate thickness is considered. The unknown field variable is described using higher order theories through the unified formulation, taking into account a general interpolation of the unknown variables with Lagrange polynomials across the physical domain. The Hamiltonian Principle is adopted for the derivation of the dynamic equilibrium equations, which arediscretized numerically by using the Generalized Differential Quadrature (GDQ) method. In addition, an efficient isogeometric NURBS-based mapping is adopted for the distortion of the physical domain. The boundary conditions of the differential problem are modelled through a general distribution of linear elastic springs along the lateral surfaces of the three-dimensional doubly-curved solid. The theory is then applied to study the vibrational modes of structures with different curvatures and lamination schemes, taking into account a general distribution of CNTs along the thickness direction. The homogenized properties of CNTs layers are obtained from the Mori-Tanaka procedure, which accounts for the agglomeration effects of the dispersed nanofibers within the matrix. Unlike previous studies focusing on doubly-curved shells reinforced with composite materials and CNTs, the present formulation enables to derive the vibration characteristics of structures made of generally anisotropic materials with general orientations. Furthermore, the calibration of the parameters for the linear elastic springs’ distribution leads to general external constraints within a single element, thus reducing the computational cost of the problem.

Analytical solutions for hydrodynamic responses of arrays of floating truncated cylinders using multi-term Galerkin method and its application to a new wave energy converter device

In the context of linear water wave theory, the analytical solutions for the diffraction and radiation of a truncated cylinder array are developed in the presence of ambient incident waves. Each cylinder in the array can oscillate with five degrees of freedom (DOFs), i.e., surge, sway, heave, roll, and pitch. This paper adopts the multi-term Galerkin method to expand the fluid velocity at the interface of different regions into a set of basis functions containing Gegenbauer polynomials, which accurately and efficiently characterizes the cube root singularity of the fluid velocity near the edges of the truncated cylinders. Using the dynamic equilibrium equations, the amplitudes of each DOF of the cylinders in the array are solved. The analytical solution presented in this paper converges rapidly, and high-precision hydrodynamic response results can be obtained using just a few truncated terms (e.g., the upper bounds of m0 = 5 and p0 = 22 can yield results of five-figure accuracy). For the 4-cylinder array, under the same accuracy conditions (the error less than 1%), the computation time of the conventional method developed by Zeng et al. [“Hydrodynamic interactions between waves and cylinder arrays of relative motions composed of truncated floating cylinders with five degrees of freedom,” J. Fluids Struct. 115, 103785 (2022d)] based on the exact algebraic method [Kagemoto and Yue, “Interactions among multiple three-dimensional bodies in water waves: An exact algebraic method,” J. Fluid Mech. 166, 189–209 (1986)] is 3.9 times longer than that of the present method. As the number of cylinders increases, the advantage of the present method in terms of convergence speed becomes more apparent, e.g., for the 16-cylinder array, the conventional solution takes 6.3 times longer than the present solution. To extract wave energy more efficiently, a new 5DOF wave energy converter (WEC) device that can extract energy in 5DOFs is proposed. The present method is adopted to investigate the hydrodynamic performance of the 5DOF WEC arrays. Compared with the traditional 1DOF (heave) WEC, the 5DOF WEC can significantly improve the energy capture performance of arrays, especially in the high-frequency wave region.

Analytical solution for simplifying the pile-soil interaction to a spring-damping system under horizontal vibration

An analytical solution is developed to investigate the model of simplifying the pile-soil interaction to a spring-damping system when the horizontal earthquake input or the top of the model is subjected to horizontal dynamic load. Based on Euler-Bernoulli theory, the dynamic equilibrium equations of pile in soil and air are established, respectively. Then, utilizing the corresponding boundary conditions, potential function decomposition, and transfer matrix method, the analytical expression of impedance at the pile head whether the horizontal earthquake input or the top of the system is subjected to horizontal dynamic load can be presented. Furthermore, the model of the pile-soil complex interaction can be simplified as a spring-damping system according to the stress equilibrium at the pile bottom in the air. Moreover, the rationality of the present solution (without simplification) is verified by comparison with existing methods, and then the difference between the present solution (without simplification) and the spring-damping system (simplification) is also compared. Finally, the influence of the parameter variation in the pile buried in the soil and soil free-field on the simplified model are discussed. The research results will be instructive for the simplification of numerical model for pile-soil interaction.

The problem of the structure and the soil plastic environment interaction in the conditions of dynamic evolutionary processes

Dynamic influences are one of the important factors affecting project decisions during the construction of buildings and structures. In the case of dynamic analysis, the range of possible variations in structural response is very difficult to determine. The complexity is determined not only by the parameters of the load itself (intensity, the law of change over time), but also by the characteristics of the research object (their values and change over time), which is an interconnected system of the structure and the soil base. Dynamic factors of influence can include periodic vibration or shock loads, the effects of explosions that cause a sharp change in pressure on the structure of the building, seismic fluctuations, etc. In many cases, the load is the result of stress waves propagating in the soil base. These factors can act on the natural environment long-term, relatively short-term, short-term and instantaneous. Therefore, the development of refined approaches for assessing the stress-strain state of structures during the interaction of the structure and the soil environment under the conditions of dynamic evolutionary processes remains an actual task today. In order to effectively find a solution to the dynamics problems, it is necessary to possess the entire set of analysis tools, to clearly present the rules and limits of their application. The ability to correctly assess the engineering and geological conditions of the construction site, the properties of the soils, the joint work of these soils with the foundations and structures of the building has always been and remains the basis of reliable and effective engineering design. The given statement of the problem is a theoretical basis to build the numerical approaches to the study of the vibrational processes of heterogeneous spatial bodies during their contact interaction with an elastic-plastic medium. The paper considers the modeling aspects of the structure and the soil elastic-plastic environment interaction under the conditions of dynamic evolutionary processes. The basic geometric and physical equations of the elasticity theory for heterogeneous circular and prismatic spatial bodies are given. In the range of solving the dynamics problem, the peculiarities of the mathematical model implementation, regarding the representation of the variation of kinetic energy as part of the equations of dynamic equilibrium of linear and nonlinear mechanical systems, are considered.

Energy extraction and optimization studies on vortex-induced vibration of the elastically mounted rigid cylinder using wake oscillator model

The vortex-induced vibration (VIV) response of an elastically mounted rigid cylinder from the standpoint of energy harvesting is analyzed with the aid of the wake oscillator model (WOM). A model of an elastically mounted rigid cylinder free to oscillate in the in-line (IL) and cross-flow (CF) directions subjected to different uniform flow velocities is considered and its responses under different mass ratios are predicted using the modified wake oscillator model. For modeling structure motion, the dynamic equilibrium equation, and for modeling the fluid part, the Van der Pol wake oscillator is utilized and they are coupled together to get the response in IL and CF directions. A comparative study is performed between the numerical and the experimental results and they are found to be in good agreement. The study is then probed into the scope of energy extraction from VIV which will presumably throw light into the development of VIV-based energy harvesting methods which is a potential future hope for the renewable energy sector. Parametric optimization study is carried out to determine the maximum harnessable energy from VIV. The response of the cylinder for the optimal parametric combination is analyzed in the time and frequency domain. It is observed that damping and reduced velocity play a crucial role in the energy extraction.

A stable numerical framework for long-time dynamic crack analysis

This paper introduces a stable numerical framework designed to address dynamic crack problems over long-time intervals. The initial step involves the temporal discretization of the governing dynamic equilibrium equations using the arbitrary order Krylov deferred correction method. To ensure precise boundary condition matching, a novel numerical implementation is incorporated into the Krylov deferred correction technique. Subsequently, the resulting system of spatial partial differential equations at each time node is solved using the meshless generalized finite difference method with 4th-order expansions in regions close to the crack-tips and 2nd-order expansions in areas far from the crack-tips. This combined approach capitalizes on the strengths of both the Krylov deferred correction technique and the generalized finite difference method, enabling stable simulations of dynamic cracks with the large time step and without the need for mesh generation. Notably, we refine the collocation nodes near crack tips to attain accurate numerical results for displacement and stress field. Several numerical experiments involving diverse impact loadings are conducted to validate the developed framework. Furthermore, a comparison is made between the dynamic stress intensity factors obtained using our approach and those from existing methods.

Study on influence coefficients of hydrodynamic force on dynamic response of deep-water pier under earthquake

The hydrodynamic force generated by the interaction between deep-water pier and water under earthquake exerts a significant influence on the dynamic response of the pier. At present, there is a lack of characterization methods to measure the influence degree of pier-water interaction on the dynamic response of the deep-water pier. Based on the structural dynamics theory and considering the coupling effect of pier-water interaction, the dynamic equilibrium equations for the deep-water pier under earthquake are established, and the solutions are deduced and validated by the finite element simulation results. Then the comprehensive analysis of influence mechanism is conducted. Further, the displacement influence coefficients and internal force influence coefficients are proposed to evaluate the influence degree of hydrodynamic force on dynamic response of deep-water piers. Validation shows that the proposed coefficients have high accuracy, indicating the important application of the proposed coefficients in seismic design of deep-water piers in real practice.

Suspension and tyre loads estimation of an FSAE car: model development and on-track validation

In this paper a relatively simple model is presented, able to compute the loads on the suspension arms and at the tyres' contact patches of a Formula SAE car starting from telemetry data. The model is based on standard dynamic equilibrium equations, a simplified assumption for the limited-slip differential and a given front-to-rear brake distribution. The model inputs are the signals acquired with common telemetry sensors, which usually equip a Formula SAE vehicle (Inertia Measurement Unit, GPS, potentiometers, Hall sensors). In order to validate the model, some arms of the suspensions were instrumented with strain gauges, and a kinematic model of the suspension, together with its static equilibrium equation set, was set up to relate the arm loads to the forces acting at the tyre-road contact patch. The suspension kinematic and equilibrium models were first validated through static tests, with known forces applied at the wheel hub. Then, the complete vehicle model was validated by comparing the same quantities, relative to some real driving tests on a kart circuit, showing a fairly good agreement between the predicted and the measured loads. The model represents a valuable tool for vehicle design and development for the Formula SAE team.

Elastodynamic Green's functions for sandwich panels with aluminum foam core and transversely isotropic face sheets using potential functions method

Green's functions of stress and displacement due to wave propagation in a sandwich panel consisting of transversely isotropic face-sheets and aluminum foam core are developed in the present paper. Dynamic partial differential equations of equilibrium for the considered face-sheets and core are expressed in cylindrical coordinates (r, θ, z), and converted into two separate equations using two unknown potential functions. Then, by writing the potential functions as Fourier series in the circumferential direction and using Hankel transform in the radial direction, an analytical solution to the potential functions in Hankel transformed space is developed. Using the boundary conditions and continuity of the problem as well as the place of application of harmonic forces, Green's functions for displacements and stresses in the frequency domain are derived using Hankel's inverse integral transform. These functions are expressed as complex integrals and calculated numerically. To verify the methodology and results, a comparison study is performed for particular cases showing a high level of the accuracy. Green's functions obtained from this paper will give information about the loading models or materials causing the maximum normal stress, shear stress, vertical and radial displacements. In addition, based on the results, with an increase in the core thickness compared to that of face-sheets in a Sandwich panel, an increment in Green's functions of stress and displacement is observed in both real and imaginary parts.

Impact Responses and Wave Dissipation Investigation of a Composite Sandwich Shell Reinforced by Multilayer Negative Poisson's Ratio Viscoelastic Polymer Material Honeycomb.

This analysis investigated the impact wave response and propagation on a composite sandwich shell when subjected to a low-velocity external shock, considering hygrothermal effects. The sandwich shell was crafted using face layers composed of functional gradient metal-ceramic matrix material and a core layer reinforced with negative Poisson's honeycomb. The honeycomb layer consisted of a combination of viscoelastic polymer material and elastic material. The equivalent parameters for the functional gradient material in the face layers were determined using the Mori-Tanaka and Voigt models, and the parameters for the negative Poisson's ratio honeycomb reinforcement core layer were obtained through Gibson's unit cell model. Parameters relevant to a low-velocity impact were derived using a modified Hertz contact law. The internal deformations, strains, and stress of the composite sandwich shell were described based on the higher-order shear deformation theory. The dynamic equilibrium equations were established using Hamilton's principle, and the Galerkin method along with the Newmark direct integration scheme was employed to calculate the shell's response to impact. The validity of the analysis was confirmed through a comparison with published literature. This investigation showed that a multilayer negative Poisson's ratio viscoelastic polymer material honeycomb-cored structure can dissipate impact wave energy swiftly and suppress shock effectively.

Analytical and numerical stability analysis of a toroidal wheel with nonholonomic constraints

In this paper, a detailed and comprehensive linear stability analysis of a rolling toroidal wheel is performed. The wheel is modeled as a rigid toroid-shaped body rolling without slipping on a horizontal surface. The nonlinear equations of motion constitute a Differential-Algebraic Equations system, given by the dynamic equilibrium equations augmented with the nonholonomic constraints, which arise from the no-slip condition. The circular steady motion and the linearized equations of motion along this relative equilibrium are obtained, for both the solid and hollow tori. The expressions of the linearized equations and the corresponding eigenvalues are derived analytically as a function of the torus aspect ratio. The variation of the stability boundary with the torus aspect ratio is shown. A comparison of the results obtained in the solid and hollow scenarios is included, and all the results are validated with the rolling hoop, which corresponds to a degenerate torus with zero aspect ratio. In the particular case of the steady straight-line rolling and spinning about a vertical diameter, which constitute limit motions of the circular steady motion, the critical rotational and angular speeds required for stabilization are obtained.

Analysis on aeroelastic stability of rectangular-planed air-supported membrane structures

This research investigates aeroelastic stability of rectangular-planed air-supported membrane structures subjected to wind actions. By employing potential theory to tackle air flows, the relations between extra aerodynamic forces and structural motions are resolved through the utilization of the boundary element method and the finite element method, capturing the complexities associated with structural geometries and wind pressure distributions. By combining these extra aerodynamic forces with structural dynamic equilibrium equations, critical instability wind velocities for various structural vibration modes are derived. The simulation results are validated through aeroelastic wind tunnel tests, demonstrating a close alignment between the numerical findings and experimental observations in terms of the critical wind velocity and the corresponding instability mode. Additionally, the study analyzes influences of structural geometric configurations, internal pressures, and membrane tensile stiffness on the critical instability wind velocities and corresponding instability modes. It is observed that lower aspect ratios, smaller structural spans, higher internal pressures, and greater membrane tensile stiffness can increase the critical instability wind velocity, corresponding to variations in critical instability mode shapes. Results obtained from this research are expected to provide guidance for the development of wind-resistant designs for air-supported membrane structures.

Seismic response analysis of super-high-rise building structures with three-layer isolation systems

This paper proposes a triple-layer isolation device based on the single-story isolation and double-layer isolation. By establishing dynamic equilibrium equations and conducting earthquake response comparative analyses on the same super-tall building structure using three different forms of isolation: single-story isolation, double-layer isolation, and triple-layer isolation, it is found that the triple-layer isolation device, which adds an isolation layer on the basis of the double-layer isolation device, exhibits many different characteristics in terms of its damping mechanism, seismic response rules, etc., due to the differences in the position and quantity of the isolation layer. Rare earthquakes are equivalent to ASCE maximum considered earthquake or annual occurrence probability of 2%. The research shows that, under rare earthquake conditions, the isolation effect of the triple-layer isolation device in the super-tall frame-shear structure is better than that of the single-story isolation and double-layer isolation devices. Particularly, as the number of floors increases, the triple-layer isolation device can significantly reduce the seismic isolation support tension and compression stresses, inter-story displacement, single-story shear force, overturning moment, and floor acceleration of the high-rise building structure, and concentrate the lateral displacement of the structure on the three isolation layers, dissipating most of the seismic input energy.

Nonlinear vibration study of a rotor blade based on a novel non-polynomial function

This work investigates the nonlinear free vibration of a turbomachinery blade, which is idealized as a rotating, conical, twisted shell of varying thickness reinforced with graphene platelets. Using the von Karman type of nonlinearity, the dynamic equilibrium equation for the twisted panel is obtained and solved by finite element method. The impact of the centrifugal force caused by rotation is taken into account, but the effect of the Coriolis force is neglected. The functionally graded twisted tapering conical shell has two alternative ways in which graphene platelets (GPLs) are reinforced in the thickness direction. The rule of mixture and the modified Halpin-Tsai model are used to compute the effective material properties of the twisted shell. The strain displacement relation is based on a new non-polynomial shear deformation theory. The parametric analysis performed at the endshows that nonlinear frequency increases when twist angle, rotational speed, tapper ratio, and weight fraction of GPLs increase. It is also observed that mode shapes are severely impacted by twist angle and rotational speed.

A multi-scale formulation for polycrystalline materials accounting for cohesive micro-cracks: Homogenisation of the traction-separation law

This contribution investigates the modelling of failure in polycrystalline materials due to the nucleation and propagation of cohesive cracks at the micro-scale. First, a two-scale formulation that enables the analysis of failure, even when the evolution and propagation of cohesive micro-cracks induce material instability, is proposed. The development of this macro-continuous/micro-discontinuous formulation is based on the Method of Multi-Scale Virtual Power, from which dynamic equilibrium and homogenisation equations are derived, ensuring a variationally consistent scale transition. The respective minimal kinematical constraint is retrieved and enforced with the Lagrange multiplier method. The formulation is also extended for periodic boundary conditions. The discretisation of the resulting equilibrium equations, with the finite element method and the generalised-α method, is described in detail.Second, a fracture-based computational homogenisation procedure is developed to obtain homogenised traction-separation laws and fracture properties from microstructural volume elements. It is derived from a crack-based Hill-Mandel principle and employs a novel energetic-based damage variable, within the Park-Paulino-Roesler cohesive model, proposed to define the crack domain and compute the crack homogenised quantities. A strategy for accurately computing the homogenised unit normal vector of the equivalent macroscopic crack is also suggested.Numerical examples demonstrate the formulation’s ability to analyse fracture problems, including a study on the impact of different microscopic boundary conditions. This work also reveals the convergence of the traction-separation law with increasing microstructure model size when using the fracture-based computational homogenisation. Furthermore, it investigates how advanced deformation mechanisms, like slip plasticity and martensitic transformation, affect intergranular crack propagation mechanisms.