Review of the dynamic stiffness method for free-vibration analysis of beams

The application of the dynamic stiffness method(DSM)for free-vibration analysis of beams is surveyed in this paper.The historical development of the DSM,which has taken place in several stages,is discussed in detail with reference to the free-vibration problems of beams.In particular,the suitability of the DSM in solving the free-vibration problems of beams through the application of the well-known Wittrick–Williams algorithm as a solution technique is highlighted.The literature concerning homogeneous isotropic metallic beams,for which the DSM is well established,is reviewed first,after which,with the rapid and ongoing emergence of advanced composite materials,the development of the DSM in solving the free-vibration problems of anisotropic beams is discussed.The free-vibration analysis of functionally graded beams using the DSM is also highlighted.The survey covers the DSM application for free-vibration analysis of a wide range of beams,including sandwich beams,rotating beams,twisted beams,moving beams and bending-torsion coupled beams,amongst others.Some aspects of the contributions made by the author and his research team are also highlighted.Finally,the future potential of the DSM in solving complex engineering problems is projected.

Force Control Compensation Method with Variable Load Stiffness and Damping of the Hydraulic Drive Unit Force Control System

Each joint of hydraulic drive quadruped robot is driven by the hydraulic drive unit（HDU）, and the contacting between the robot foot end and the ground is complex and variable, which increases the difficulty of force control inevitably. In the recent years, although many scholars researched some control methods such as disturbance rejection control, parameter self-adaptive control, impedance control and so on, to improve the force control performance of HDU, the robustness of the force control still needs improving. Therefore, how to simulate the complex and variable load characteristics of the environment structure and how to ensure HDU having excellent force control performance with the complex and variable load characteristics are key issues to be solved in this paper. The force control system mathematic model of HDU is established by the mechanism modeling method, and the theoretical models of a novel force control compensation method and a load characteristics simulation method under different environment structures are derived, considering the dynamic characteristics of the load stiffness and the load damping under different environment structures. Then, simulation effects of the variable load stiffness and load damping under the step and sinusoidal load force are analyzed experimentally on the HDU force control performance test platform, which provides the foundation for the force control compensation experiment research. In addition, the optimized PID control parameters are designed to make the HDU have better force control performance with suitable load stiffness and load damping, under which the force control compensation method is introduced, and the robustness of the force control system with several constant load characteristics and the variable load characteristics respectively are comparatively analyzed by experiment. The research results indicate that if the load characteristics are known, the force control compensation method presented in this paper has positive compensation effects on the load characteristics variation, i.e., this method decreases the effects of the load characteristics variation on the force control performance and enhances the force control system robustness with the constant PID parameters, thereby, the online PID parameters tuning control method which is complex needs not be adopted. All the above research provides theoretical and experimental foundation for the force control method of the quadruped robot joints with high robustness.

Dynamic stiffness for thin-walled structures by power series

The dynamic stiffness method is introduced to analyze thin-walled structures including thin-walled straight beams and spatial twisted helix beam. A dynamic stiffness matrix is formed by using frequency dependent shape functions which are exact solutions of the governing differential equations. With the obtained thin-walled beam dynamic stiffness matrices, the thin-walled frame dynamic stiffness matrix can also be formulated by satisfying the required displacements compatibility and forces equilib-rium, a method which is similar to the finite element method (FEM). Then the thin-walled structure natural frequencies can be found by equating the determinant of the system dynamic stiffness matrix to zero. By this way, just one element and several elements can exactly predict many modes of a thin-walled beam and a spatial thin-walled frame, respectively. Several cases are studied and the results are compared with the existing solutions of other methods. The natural frequencies and buckling loads of these thin-walled structures are computed.

Dynamic characteristics of a novel damped outrigger system

This paper presents exact analytical solutions for a novel damped outrigger system, in which viscous dampers are vertically installed between perimeter columns and the core of a high-rise building. An improved analytical model is developed by modeling the effect of the damped outrigger as a general rotational spring acting on a Bernoulli-Euler beam. The equivalent rotational spring stiffness incorporating the combined effects of dampers and axial stiffness of perimeter columns is derived. The dynamic stiffness method(DSM) is applied to formulate the governing equation of the damped outrigger system. The accuracy and effi ciency are verifi ed in comparison with those obtained from compatibility equations and boundary equations. Parametric analysis of three non-dimensional factors is conducted to evaluate the infl uences of various factors, such as the stiffness ratio of the core to the beam, position of the damped outrigger, and the installed damping coeffi cient. Results show that the modal damping ratio is signifi cantly infl uenced by the stiffness ratio of the core to the column, and is more sensitive to damping than the position of the damped outrigger. The proposed analytical model in combination with DSM can be extended to the study of structures with more outriggers.

In-plane dynamic Green's functions for inclined and uniformly distributed loads in a multi-layered transversely isotropic half-space

The dynamic stiffness method combined with the Fourier transform is utilized to derive the in-plane Green’s functions for inclined and uniformly distributed loads in a multi-layered transversely isotropic（TI）half-space.The loaded layer is fixed to obtain solutions restricted in it and the corresponding reactions forces,which are then applied to the total system with the opposite sign.By adding solutions restricted in the loaded layer to solutions from the reaction forces,the global solutions in the wavenumber domain are obtained,and the dynamic Green’s functions in the space domain are recovered by the inverse Fourier transform.The presented formulations can be reduced to the isotropic case developed by Wolf（1985）,and are further verified by comparisons with existing solutions in a uniform isotropic as well as a layered TI halfspace subjected to horizontally distributed loads which are special cases of the more general problem addressed.The deduced Green’s functions,in conjunction with boundary element methods,will lead to significant advances in the investigation of a variety of wave scattering,wave radiation and soil-structure interaction problems in a layered TI site.Selected numerical results are given to investigate the influence of material anisotropy,frequency of excitation,inclination angle and layered on the responses of displacement and stress,and some conclusions are drawn.

A Theoretical Study the Effect of Openings in Infill Wall Panels on the Behavior of Structures

Two dimensional,reinforced concrete building frames built on raft foundation and having infill wall panels with openings in them are analysed using the direct stiffness method.Beams and columns are modelled by beam column elements.Wall panels are modelled by plane stress finite elements.The raft foundation is modelled by uniaxial finite elements.The soil is modelled as half space model.Openings in wall panels are introduced by using fictitious beams between real floor beams. A computer program is written to carry out the static analysis and do the necessary comparison to show the effect of openings on the structural behavior.

Poroelastoplastic reservoir modeling by tangent stiffness matrix method

As oil and gas extraction activities move into deeper rock formations,many experimental studies and field investigations indicate rock exhibits a plastic behavior rather than a pure linear elastic behavior,so poroelastoplasticity must be taken into account in the reservoir simulation.Because reservoir rock is a porous material consisting of a compressible solid matrix and number of compressible fluids occupying the pore space,fully coupled modeling is required for reservoir simulation considering solid-fluid interaction,complex stress conditions and nonlinear behaviors.But the computational process could be cumbersome when constant tangent stiffness method is used to address the poroelastoplastic behavior.In this paper,a fully coupled poroelastoplasticity reservoir model based on Drucker-Prager yield criterion is implemented the tangent stiffness method,and the computational efficiency is compared with the constant stiffness method.The accuracy of these two methods is demonstrated in one-dimensional consolidation.In a case study,these two methods are used to analyze the stresses and pore pressure of a reservoir and computing results and running efficiency are compared.Also,the linear elastic and nonlinear solutions are compared in one-dimensional consolidation and reservoir modeling.It shows that the difference between results by constant stiffness method and tangent stiffness method is very small,while the tangent stiffness method shows significantly fewer iteration numbers and shorter running time than the constant stiffness method.

FEM and ANN analyses of short and tall geosynthetic reinforced soil walls:Evaluation of various effective aspects and ASSHTO acceptable design methods

In addition to confined investigations on tall geosynthetic reinforced soil(GRS)walls,a remarkable database of such walls must be analyzed to diminish engineers’concerns regarding the American Association of State Highway and Transportation Officials(AASHTO)Simplified or Simplified Stiffness Method in projects.There are also uncertainties regarding reinforcement load distributions of GRS walls at the connections.Hence,the current study has implemented a combination of finite element method(FEM)and artificial neural network(ANN)to distinguish the performance of short and tall GRS walls and assess the AASHTO design methods based on 88 FEM and 10000 ANN models.There were conspicuous differences between the effectiveness of stiffness(63%),vertical spacing(22%),and length of reinforcements(14%)in the behavior of short and tall walls,along with predictions of geogrid load distributions.These differences illustrated that using the Simplified Method may exert profound repercussions because it does not consider wall height.Furthermore,the Simplified Stiffness Method(which incorporates wall height)predicted the reinforcement load distributions at backfill and connections well.Moreover,a Multilayer Perceptron(MLP)algorithm with a low average overall relative error(up to 2.8%)was developed to propose upper and lower limits of reinforcement load distributions,either at backfill or connections,based on 990000 ANN predictions.

Propagation behavior of elastic waves in one-dimensional piezoelectric/piezomagnetic phononic crystal with line defects

Using a stiffness matrix method, we in- vestigate the propagation behaviors of elastic waves in one-dimensional （1D） piezoelectric/piezomagnetic （PE/PM） phononic crystals （PCs） with line defects by calculating energy reflection/transmittion coefficients of quasi-pressure and quasi-shear waves. Line defects are created by the re- placement of PE or PM constituent layer. The defect modes existing in the first gap are considered and the influences on defect modes of the material properties and volume fraction of the defect layers, the type of incident waves, the location of defect layer and the number of structural layers are discussed in detail. Numerical results indicate that defect modes are the most obvious when the defect layers are inserted in the middle of the perfect PCs; the types of incidence wave and material properties of the defect layers have important effects on the numbers, the location of frequencies and the peaks of defect modes, and the defect modes are strongly de- pendent on volume fraction of the defect layers. We hope this paper will be found useful for the design of PE/PM acoustic filters or acoustic transducer with PCs structures.

On the transient models of the VITAS code:applications to the C5G7-TD pin-resolved benchmark problem

This article describes the transient models of the neutronics code VITAS that are used for solving time-dependent,pinresolved neutron transport equations.VITAS uses the stiffness confinement method(SCM)for temporal discretization to transform the transient equation into the corresponding transient eigenvalue problem(TEVP).To solve the pin-resolved TEVP,VITAS uses a heterogeneous variational nodal method(VNM).The spatial flux is approximated at each Cartesian node using finite elements in the x-y plane and orthogonal polynomials along the z-axis.Angular discretization utilizes the even-parity integral approach at the nodes and spherical harmonic expansions at the interfaces.To further lower the computational cost,a predictor–corrector quasi-static SCM(PCQ-SCM)was developed.Within the VNM framework,computational models for the adjoint neutron flux and kinetic parameters are presented.The direct-SCM and PCQ-SCM were implemented in VITAS and verified using the two-dimensional(2D)and three-dimensional(3D)exercises on the OECD/NEA C5G7-TD benchmark.In the 2D and 3D problems,the discrepancy between the direct-SCM solver’s results and those reported by MPACT and PANDAS-MOC was under 0.97%and 1.57%,respectively.In addition,numerical studies comparing the PCQ-SCM solver to the direct-SCM solver demonstrated that the PCQ-SCM enabled substantially larger time steps,thereby reducing the computational cost 100-fold,without compromising numerical accuracy.

A numerical framework for underground structures in layered ground under inclined P-SV waves using stiffness matrix and domain reduction methods

A numerical framework was proposed for the seismic analysis of underground structures in layered ground under inclined P-SV waves.The free-field responses are first obtained using the stiffness matrix method based on plane-wave assumptions.Then,the domain reduction method was employed to reproduce the wavefield in the numerical model of the soil–structure system.The proposed numerical framework was verified by providing comparisons with analytical solutions for cases involving free-field responses of homogeneous ground,layered ground,and pressure-dependent heterogeneous ground,as well as for an example of a soil–structure interaction simulation.Compared with the viscous and viscous-spring boundary methods adopted in previous studies,the proposed framework exhibits the advantage of incorporating oblique incident waves in a nonlinear heterogeneous ground.Numerical results show that SV-waves are more destructive to underground structures than P-waves,and the responses of underground structures are significantly affected by the incident angles.

TWO ITERATION METHODS FOR SOLVING LINEAR ALGEBRAIC SYSTEMS WITH LOW ORDER MATRIX A AND HIGH ORDER MATRIX B:Y=(AB)Y+Ф

This paper presents optimum an one-parameter iteration (OOPI) method and a multi-parameter iteration direct (MPID) method for efficiently solving linear algebraic systems with low order matrix A and high order matrix B: Y = (A B)Y + Φ. On parallel computers (also on serial computer) the former will be efficient, even very efficient under certain conditions, the latter will be universally very efficient.

A numerical model for pipelaying on nonlinear soil stiffness seabed

The J-lay method is regarded as one of the most feasible methods to lay a pipeline in deep water and ultra-deep water. A numerical model that accounts for the nonlinear soil stiffness is developed in this study to evaluate a J-lay pipeline. The pipeline considered in this model is divided into two parts： the part one is suspended in water, and the part two is laid on the seabed. In addition to the boundary conditions at the two end points of the pipeline, a special set of the boundary conditions is required at the touchdown point that connects the two parts of the pipeline. The two parts of the pipeline are solved by a numerical iterative method and the finite difference method, respectively. The proposed numerical model is validated for a special case using a catenary model and a numerical model with linear soil stiffness. A good agreement in the pipeline configuration, the tension force and the bending moment is obtained among these three models. Furthermore, the present model is used to study the importance of the nonlinear soil stiffness. Finally, the parametric study is performed to study the effect of the mudline shear strength, the gradient of the soil shear strength, and the outer diameter of the pipeline on the pipelaying solution.

Discrete element modeling of the effect of particle size distribution on the small strain stiffness of granular soils

Discrete element modeling was used to investigate the effect of particle size distribution on the small strain shear stiffness of granular soils and explore the fundamental mechanism controlling this small strain shear stiffness at the particle level. The results indicate that the mean particle size has a negligible effect on the small strain shear modulus. The observed increase of the shear modulus with increasing particle size is caused by a scale effect. It is suggested that the ratio of sample size to the mean particle size should be larger than 11.5 to avoid this possible scale effect. At the same confining pressure and void ratio, the small strain shear modulus decreases as the coefficient of uniformity of the soil increases. The Poisson＇s ratio decreases with decreasing void ratio and increasing confining pressure instead of being constant as is commonly assumed. Microscopic analyses indicate that the small strain shear stiffness and Poisson＇s ratio depend uniquely on the soil＇s coordination number.

GRID-INDEPENDENT CONSTRUCTION OF MULTISTEP METHODS

A new polynomial formulation of variable step size linear multistep methods is pre- sented, where each k-step method is characterized by a fixed set of k - 1 or k parameters. This construction includes all methods of maximal order （p = k for stiff, and p = k ＋ 1 for nonstiff problems）. Supporting time step adaptivity by construction, the new formulation is not based on extending classical fixed step size methods; instead classical methods are obtained as fixed step size restrictions within a unified framework. The methods are imple- mented in MATLAB, with local error estimation and a wide range of step size controllers. This provides a platform for investigating and comparing different multistep method in realistic operational conditions. Computational experiments show that the new multi- step method construction and implementation compares favorably to existing software, although variable order has not yet been included.

Stability Analysis of a Heterogeneous Periodic Fluid-Conveying Nanotube System

In this paper,the stability of a periodic heterogeneous nanotube conveying fluid is investigated.The governing equations of the nanotube system are derived based on the nonlocal Euler-Bernoulli beam theory.The dynamic stiffness method is employed to analyze the natural frequencies and critical flow velocities of the heteronanotube.The results and discussions are presented from three aspects which reveal the influences of period number,material length ratio and boundary conditions.In particular,we make comparisons between the heterogeneous nanotubes with periodic structure and the homogeneous ones with the same integral values of material properties along the longitudinal direction to isolate the influences of periodic distribution.According to the simulation results,we can conclude that with a proper selection of period number in terms of length ratio,the stability of the constructed nanotube can be improved.

Crack identification in functionally graded material framed structures using stationary wavelet transform and neural network

In this paper,an integrated procedure is proposed to identify cracks in a portal framed structure made of functionally graded material(FGM)using stationary wavelet transform(SWT)and neural network(NN).Material properties of the structure vary along the thickness of beam elements by the power law of volumn distribution.Cracks are assumed to be open and are modeled by double massless springs with stiffness calculated from their depth.The dynamic stiffness method(DSM)is developed to calculate the mode shapes of a cracked frame structure based on shape functions obtained as a general solution of vibration in multiple cracked FGM Timoshenko beams.The SWT of mode shapes is examined for localization of potential cracks in the frame structure and utilized as the input data of NN for crack depth identification.The integrated procedure proposed is shown to be very effective for accurately assessing crack locations and depths in FGM structures,even with noisy measured mode shapes and a limited amount of measured data.

Investigation into improving the efficiency and accuracy of CFD/DEM simulations

The Euler-Lagrange approach combined with a discrete element method has frequently been applied to elucidate the hydrodynamic behavior of dense fluid-solid flows in fluidized beds. In this work, the efficiency and accuracy of this model are investigated. Parameter studies are performed; in these studies, the stiffness coefficient, the fluid time step and the processor number are varied under conditions with different numbers of particles and different particle diameters. The obtained results are compared with measurements to derive the optimum parameters for CFD/DEM simulations. The results suggest that the application of higher stiffness coefficients slightly improves the simulation accuracy. However, the average computing time increases exponentially. At larger fluid time steps, the results show that the average computation time is independent of the applied fluid time step whereas the simulation accuracy decreases greatly with increasing the fluid time step. The use of smaller time steps leads to negligible improvements in the simulation accuracy but results in an exponential rise in the average computing time. The parallelization accelerates the DEM simulations if the critical number for the domain decomposition is not reached. Above this number, the performance is no longer proportional to the number of processors. The critical number for the domain decomposition depends on the number of particles. An increase in solid contents results in a shift of the critical decomposition number to higher numbers of CPUs.

Stifness modeling and analysis of a novel 4-DOF PKM for manufacturing large components

Faster response to orientation varying is one of the outstanding abilities of a parallel kinematic machine（PKM）.It enables such a system to act as a reconfgurable module employed to machine large components effciently.The stiffness formulation and analysis are the beforehand key tasks for its parameters design.A novel PKM with four degrees of freedom（DOFs）is proposed in this paper.The topology behind it is 2PUS-2PRS parallel mechanism.Its semianalytical stiffness model is frstly obtained,where the generalized Jacobian matrix of 2PUS-2PRS is formulated with the help of the screw theory and the stiffness coeffcients of complicated components are estimated by integrating fnite element analysis and numerical ftting.Under the help of the model,it is predicted that the property of system stiffness distributes within the given workspace,which features symmetry about a certain plane and is also verifed by performing fnite element analysis of the virtual prototype.Furthermore,key parameters affecting the system stiffness are identifed through sensitivity analysis.These provide insights for further optimization design of this PKM.

Static response of a layered magneto-electro-elastic half-space structure under circular surface loading

A cylindrical system of vector functions, the stiffness matrix method and the corresponding recursive algorithm are proposed to investigate the static response of transversely isotropic,layered magneto-electro-elastic（MEE） structures over a homogeneous half-space substrate subjected to circular surface loading. In terms of the system of vector functions, we expand the extended displacements and stresses, and deduce two sets of ordinary differential equations, which are related to the expansion coeficients. The solution to one of the two sets of these ordinary differential equations can be evaluated by using the stiffness matrix method and the corresponding recursive algorithm. These expansion coeficients are then integrated by adaptive Gaussian quadrature to obtain the displacements and stresses in the physical domain. Two types of surface loads, mechanical pressure and electric loading,are considered in the numerical examples. The calculated results show that the proposed technique is stable and effective in analyzing the layered half-space MEE structures under surface loading.