Direct Numerical Integration Method Research Articles

Series–Parallel System Analysis and Reliability Research of Grid Structures Considering Adaptive Dynamic Bounding Threshold

There are many bars in grid structures, and their topological relationships are complex, so the workload of calculations involved in researching their reliability is large. In this paper, based on the geometric topological relationship within a grid structure, a series–parallel system analysis of the structure is carried out by means of node analysis, constraint grade division, and hierarchical combination analysis, and the series–parallel analysis results for the structure are used to judge the failure of the structure. In order to avoid the problem of missing important components due to unreasonable selection of the bounding threshold when identifying failed components, an adaptive dynamic bounding threshold method with high accuracy is introduced. The failure probability value of the failure path is calculated using the direct numerical integration method, and the reliability of the whole structure is obtained. Finally, the validity of this method is verified using an example of a square pyramid grid structure. The results show that the method is feasible for series–parallel system analysis, structural failure judgment, and reliability analysis of a grid structure.

Fuzzy Neural Network PID Control Used in Individual Blade Control

In order to further reduce the vibration level of helicopters, the active vibration control technology of helicopters has been extensively studied. Among them, individual blade control (IBC) independently applies high-order harmonics to each blade with an actuator, which can improve the aerodynamic environment of the blade and effectively reduce the vibration load of the hub. The rotor structural dynamics model based on the Hamilton energy variation principle and the medium deformation beam theory were established firstly, and the aerodynamic model based on the dynamic inflow model and the Leishman–Beddoes unsteady aerodynamic model were also established. The structural finite element method and the direct numerical integration method were used to calculate the vibration response of the rotor to determine the vibration load of the hub. After these, the steepest descent-golden section combinatorial optimization algorithm was used to find the optimization parameters of IBC. Based on this, the input parameters of fuzzy neural network PID control were determined, and the rotor hub vibration load control simulation was conducted. Under the effect of IBC, the vibration loads of the hub could be reduced by about 60%. The article gives the best control laws of individual harmonic pitch control and their combinations. These results can theoretically be applied to the design of control law to reduce helicopter vibration loads.

An efficient method for predicting wheel-rail forces in coupled nonlinear train-track-bridge system using artificial neural networks

Wheel-rail forces are an essential indicator for vehicle safety evaluation. The calculation of wheel-rail forces for the coupled nonlinear train-track-bridge system using the direct numerical integration method is time-consuming, hindering the timely safety and reliability assessment of train operations. In this paper, an efficient method based on the nonlinear autoregressive with exogenous input artificial neural network (NARX-ANN) was proposed to predict the nonlinear wheel-rail force of trains on bridges caused by track irregularity. Firstly, samples of track irregularity time history were randomly generated using the stochastic harmonic function method. The vertical and lateral wheel-rail forces of the coupled nonlinear train-track-bridge system were calculated by numerical integration. Secondly, the NARX-ANN model was established and trained with a small number of randomly selected wheel-rail force samples. Finally, the wheel-rail forces obtained by numerical integration were regarded as the ground truth to verify the prediction accuracy of the NARX-ANN model. The influence of training configuration, initial output, and the number of time delays on prediction accuracy was systematically analyzed. The results showed that the NARX-ANN model could accurately predict the time history of wheel-rail forces. In addition, its computational efficiency is 19.35 times higher than that of the numerical integration method. It is hoped that this study can guide the stochastic analysis of the train–track–bridge system.

A novel lever-type elastic metamaterial model for low-frequency wave attenuation

Elastic metamaterials have received extensive attention due to their proven track record in wave attenuation. In this study, a novel one-dimensional lever-type elastic metamaterial model is designed to achieve low-frequency wave attenuation by introducing an adjustable lever mechanism in a conventional mass-spring resonator. The dispersion relation and effective mass expression of the proposed metamaterial are theoretically derived. Then, the band structure characteristics and the effect of parameters on the band gap are analyzed in detail. The results show that by selecting appropriate lever ratio and initial rotation angle, a low frequency band gap can be obtained and the wave attenuation capability within the band gap can be enhanced. Finally, the dynamic response of the proposed structure is calculated by direct numerical integration method to verify the analytical results. Furthermore, the application potential of the proposed elastic metamaterial in shock wave attenuation is demonstrated by analyzing the spatial propagation of wave packets along the proposed structure. This study provides a meaningful reference for the design of other tunable elastic metamaterials to attenuate waves.

Probabilistic seismic performance evaluation of nuclear containment structure subjected to far-fault ground motions

This study presents probabilistic seismic performance evaluation of nuclear containment structure subjected to far-fault ground motions. To achieve this objective, refined finite element model of nuclear containment structure is established and incremental dynamic analysis (IDA) is performed automatically using pre-processing and post-processing scripts. Cracking status, roof drift, damage dissipated energy and floor acceleration of the nuclear containment structure is discussed. Furthermore, probabilistic seismic safety margin of nuclear containment structure is quantitatively evaluated by median value method and confidence interval method. Finally, a closed-form solution for the annual frequency of exceedance for the roof drift of the nuclear containment structure is derived, and the effectiveness of the closed-form solution is verified by the classical direct numerical integration method. Annual frequency of exceedance for the roof drift of the nuclear containment structure corresponding to different performance levels is analyzed. Results indicated that the bottom of nuclear containment structure is the most vulnerable location. In general, nuclear containment structure subjected to far-fault ground motions is dominated by shearing cracks. When the ground motion intensity level is less than 0.4 g, damage dissipated energy of the nuclear containment structure is very small. In general, with the increase of elevation, acceleration amplification factor of the nuclear containment structure increases rapidly. Probabilistic seismic safety margin of the nuclear containment structure evaluated by median value method tends to be conservative. Analytical and classical numerical integration method provides almost the same prediction of the annual frequency of exceedance for the roof drift of the nuclear containment structure, which indicates the effectiveness of the proposed closed-form solution.

Beam propagation simulations for LISA in the presence of telescope aberrations

Accurate simulation of the propagation of light between the spacecraft of the laser interferometer space antenna (LISA) gravitational wave observatory will be a vital tool in determining the optical design of the telescopes used in the constellation. In this work, we examine the methods available for numerical simulation of this propagation, and consider the effect of an aberrated transmitting telescope (Tx) on the light collected by the receiving telescope (Rx). Propagation software has been developed using direct numerical integration methods, and has been validated by comparison to analytical solutions for particular cases. Zernike modal aberrations up to and including primary spherical have been considered in the Tx, and, in particular, the effects of defocus, astigmatism and coma were examined. It was found that minimization of the even radial order aberrations in Tx resulted in a reduced wavefront error at Rx, while odd aberrations such as coma can displace the maximum irradiance away from the optical axis. Thus careful consideration of the impact of telescope aberrations will be required to minimise detrimental effects on the detection of gravitational waves.

Nonlinear free vibration analysis of functionally graded beams by using different shear deformation theories

This study investigates the nonlinear free vibration of functionally graded material (FGM) beams by different shear deformation theories. The volume fractions of the material constituents and effective material properties are assumed to be changing in the thickness direction according to the power-law form. The von Kármán geometric nonlinearity has been considered in the formulation. The Ritz method and Lagrange equation are adopted to yield the discrete formulations. A direct numerical integration method for the motion equation in matrix form is developed to solve the nonlinear frequencies of FGM beams. Comparing with the global concordant deformation assumption (GCDA), a new deformation assumption named as local concordant deformation assumption (LCDA) is proposed in this study. The LCDA fits with the real deformation of the vibrating beam better, thus more accurate results of the nonlinear frequency can be expected. In numerical results, the comparison study of the GCDA and LCDA is carried out. In addition, the effects of power-law index, slenderness ratio and maximum deflection for different shear deformation theories and boundary conditions on the nonlinear frequency of the beam are discussed.

Modeling and dynamic analysis of a vehicle-flexible pavement coupled system subjected to road surface excitation

Increased road traffic combined with heavy vehicle loads leads to deterioration of pavements and reduces the life span of the paved roads. As a result, large amounts of financial resources are spent each year to improve and maintain road infrastructure around the world. Vehicle dynamics and pavement dynamics are strongly coupled through their contact points. This research focuses on the dynamic analysis of pavement-vehicle interaction system and the effect of coupling action on the response. The system response due to the moving vehicular load on rough road supported by a linear visco-elastic foundation was investigated. The vehicle is modeled as a two-degree-of-freedom quarter-vehicle model, and the pavement-foundation system is described by a simply supported Euler-Bernoulli beam resting on Pasternak foundation, while the tire is coupled to the flexible pavement with a single point contact. Galerkin method was used to develop a system of governing differential equations for a coupled system in the time domain. Direct numerical integration method using Newmark-β based on linear average acceleration method was then used to solve the governing equations and evaluate the response of the coupled system. The results were validated with previous research works and compared with conventional uncoupled systems. Finally, the effects of parameters such as vehicle speed, road roughness, soil stiffness and suspension damping on the responses were investigated.

A numerical algorithm for solving a two-layered composite beam subjected to vibrational loads

In this work, double-layer beams are being researched using Rzhanitsyn’s theory of composite rods. One of the fundamental propositions of this theory is that multiple layers of such bars are connected by absolutely rigid cross links and elastically-pliable shear links. The concentrated forces and partially-constant distributed loads which are changing in time according to the harmonious law are taken into account as the external influence. The method of direct numerical integration of the equations of motion in time is used to solve the problem of the forced oscillations of the beam. The difference equations of the serial approximations method are used to approximate partial spatial coordinate derivatives. The results of the calculation of the beam under the vibrational influence got with different intervals of spatial and temporal coordinates have been given as the illustration of the suggested method.Above-stated method might interest the specialists in the field of the analysis of building constructions because of the universality of Rzhanitsyn’s theory of built-up bars which finds application in calculation of multilayer wooden constructions, perforated metal beams, multilayer reinforced-concrete beams with different elasticity modulus, in modelling the crack formation processes, in calculation of multi-storey buildings’ frames.

Evaluation of radiation loading on finite cylindrical shells using the fast Fourier transform: A comparison with direct numerical integration.

The radiation loading on a vibratory finite cylindrical shell is conventionally evaluated through the direct numerical integration (DNI) method. An alternative strategy via the fast Fourier transform algorithm is put forward in this work based on the general expression of radiation impedance. To check the feasibility and efficiency of the proposed method, a comparison with DNI is presented through numerical cases. The results obtained using the present method agree well with those calculated by DNI. More importantly, the proposed calculating strategy can significantly save the time cost compared with the conventional approach of straightforward numerical integration.

Impact adding bifurcation in an autonomous hybrid dynamical model of church bell

Abstract In this paper we present the bifurcation analysis of the yoke-bell-clapper system which corresponds to the biggest bell “Serce Lodzi” mounted in the Cathedral Basilica of St Stanislaus Kostka, Lodz, Poland. The mathematical model of the system considered in this work has been derived and verified based on measurements of dynamics of the real bell. We perform numerical analysis both by direct numerical integration and path-following method using toolbox ABESPOL (Chong, 2016). By introducing the active yoke the position of the bell-clapper system with respect to the yoke axis of rotation can be easily changed and it can be used to probe the system dynamics. We found a wide variety of periodic and non-periodic solutions, and examined the ranges of coexistence of solutions and transitions between them via different types of bifurcations. Finally, a new type of bifurcation induced by a grazing event - an “impact adding bifurcation” has been proposed. When it occurs, the number of impacts between the bell and the clapper is increasing while the period of the system’s motion stays the same.

Evaluation of dynamic characteristics in a transmission line latticed steel tower

This paper presents a numerical dynamic analysis of a self-supporting transmission line (TL) latticed steel tower, for the purpose of compare the results with those obtained in an experimental dynamic analysis of a full-scale tower. The main objective of this work is to calibrate the structural damping (c) in the numerical model until the damping ratio (ζ) results the same that in the experimental analysis, since the damping is a fundamental parameter in numerical analysis, which can change significantly the response of the structure, especially in terms of dynamic amplification. It should be noted that in this type of structure is crucial perform a structural analysis most accurate possible, given the fact that in a TL are built hundreds of identical towers and the collapse of one can lead to a cascading failure. The dynamic characteristics, such as the frequencies and normal modes were evaluated in a modal analysis, using the ANSYS© program v. 15.0, and also through the direct numerical integration method, using an explicit approach (central finite differences). The experimental tower was tested using impulsive loads, which were reproduced in the numerical model to allow comparison with the experimental results. In the modal analysis were employed models commonly used in actual designs, that is, with space truss and space frame members. For the direct integration method, was used a program developed in FORTRAN, in which only space truss elements are allowed, necessitating the adoption of the fictitious bars to eliminate internal instabilities. The consideration of the damping in the direct integration method was taken in a simplest form, that is, proportional to the mass. Finally, the numerical values found were very close to the experimental response, setting the value of structural damping for use in such projects of this type of tower.

Integrated likelihood computation methods

Suppose a model has parameter $$\theta =(\psi , \lambda )$$ź=(ź,ź), where $$\psi $$ź is the parameter of interest and $$\lambda $$ź is a nuisance parameter. The integrated likelihood method eliminates $$\lambda $$ź from the likelihood function $$L(\psi , \lambda )$$L(ź,ź) by integrating with respect to a weight function $$\pi (\lambda | \psi )$$ź(ź|ź). The resulting integrated likelihood function $$\bar{L}(\psi )$$L¯(ź) can be used for inference for $$\psi $$ź. However, the analytical form for the integrated likelihood is not always available. This paper discusses 12 different approaches to computing the integrated likelihood. Some methods were originally developed for other computation purposes and they are modified to fit in the integrated likelihood framework. Methods considered include direct numerical integration methods such as Monte Carlo integration method, importance sampling, Laplace method; marginal likelihood computation methods; and methods for computing the marginal posterior density. Simulation studies and real data example are presented to evaluate and compare these methods empirically.

Magnetizing Curve Estimation of Induction Motors in Single-Phase Magnetization Mode Considering Differential Inductance Effect

To obtain an accurate magnetizing curve of induction motors at standstill, including both field-weakening region and saturated region, a novel method for estimation of mutual inductance $L_{m}$ is proposed. At first, the reasons causing error of $L_{m}$ identified by conventional ac-current-injection-based method, i.e., fluctuation of static $L_{m}$ and differential inductance effect, are clarified in a theoretical view. To solve the problem, a continuous ac excitation of customized low frequency, with dc offset, is designed to excite the motor in single-phase mode. While dc offset with configurable amplitudes is used to fix the saturation level and restrain variation of static $L_{m}$ , ac excitation maintains identifiability of the inductance. After the estimated $L_{m}$ is retrieved from stator-side reactance, a direct numerical-integration method is proposed to effectively correct the differential inductance effect. The influence of injection frequency is rigorously investigated, which is also a key point for successful estimation. Both simulation and experiments have been implemented to demonstrate the feasibility and validity of the proposed method.

A direct numerical integration (DNI) method to obtain wall thermal response factors

The solution of the problem of transient conduction in multi-layer walls through response factor method is well-known and has been studied widely and for a long time, and recent studies show that research in this field is not closed and keeps improving calculation methods of this fundamental issue in building energy performance.The aim of this work is to present an alternative to classical methods of inversion avoiding the process of root finding and approximating directly the integral Laplace's inversion formula. Previous attempts have been developed recently in this spirit, and a comparison of these methods with the one proposed will be shown in the results section.The numerical inversion considered in this work computes directly the inversion integral, using a quadrature rule, reason why this new method has been named direct numerical integration (DNI). It uses a specific integration path which makes the error diminish exponentially with the number of approximation points.The case studies will show the accuracy of the obtained results with very small computational effort and high programming simplicity.

Calculation of the Radiative Heat-Exchange Areas in a Large-Scale Furnace with the Use of the Monte Carlo Method

An improved approach to the simulation of the radiative heat exchange in a furnace is proposed. This approach is realized with the use of the computer program REFORM based on the well-known Monte Carlo ray tracing algorithms. The method proposed allows one to determine the factors of radiative heat exchange in furnaces for which the method of direct numerical integration is difficult to use due to their geometries. The indicated REFORM program was validated and the results obtained with it were compared with corresponding existing solutions. It has been established that the approach proposed makes it possible to more accurately represent the radiative heat exchange between a steel load and its surroundings.

Design and performance analysis of a nanogyroscope based on electrostatic actuation and capacitive sensing

In this paper, a vibrating beam gyroscope with high operational frequencies at mode-matched condition is proposed. The model comprises a micro-cantilever with attached tip mass operating in the flextural–flextural mode. The drive mode is actuated via the electrostatic force, and due to the angular rotation of the base about the longitudinal axis. The secondary sub-nanometric vibration is induced in the sense direction which causes a capacitive change in the sense electrodes. The coupled electro-mechanical equation of motion is derived using the extended Hamilton's principle, and it is solved by direct numerical integration method. The gyroscope performance is investigated through the simulation results, where the device dynamic response, rate sensitivity, resolution, bandwidth, dynamic range, g sensitivity and shock resistance are studied. The obtained results show that the proposed device may have better performance compared to commercial micro electromechanical gyroscope characteristics.

Stability and torsional vibration analysis of a micro-shaft subjected to an electrostatic parametric excitation using variational iteration method

In this article stability and parametrically excited oscillations of a two stage micro-shaft located in a Newtonian fluid with arrayed electrostatic actuation system is investigated. The static stability of the system is studied and the fixed points of the micro-shaft are determined and the global stability of the fixed points is studied by plotting the micro-shaft phase diagrams for different initial conditions. Subsequently the governing equation of motion is linearized about static equilibrium situation using calculus of variation theory and discretized using the Galerkin’s method. Then the system is modeled as a single-degree-of-freedom model and a Mathieu type equation is obtained. The Variational Iteration Method (VIM) is used as an asymptotic analytical method to obtain approximate solutions for parametric equation and the stable and unstable regions are evaluated. The results show that using a parametric excitation with an appropriate frequency and amplitude the system can be stabilized in the vicinity of the pitch fork bifurcation point. The time history and phase diagrams of the system are plotted for certain values of initial conditions and parameter values. Asymptotic analytically obtained results are verified by using direct numerical integration method.

Bifurcation analysis on a turning system with large and state-dependent time delay

Stability and bifurcation analyses were performed in this study on the turning process with a state-dependent and large time delay using the method of multiple scales (MMS). The turning system tool was modeled as an oscillator with two degrees of freedom, and both the cubic nonlinear stiffness and the nonlinear cutting force were considered. The nonlinear cutting force was appropriately expanded in a Taylor series considering the state-dependency of the time delay. The time delay and parameters were scaled through the proper ordering process to reflect the large delay effect on an asymptotic formulation of the MMS. Asymptotic solutions were then obtained by the MMS in the large delay regime and used to calculate the linear stability boundaries (i.e., Hopf bifurcation points) and coexisting one-period periodic solutions (i.e., limit cycles) of the turning system. To investigate the local and global behaviors of the tool chatter, bifurcation diagrams were obtained at various workpiece rotating speeds. The validity of the results was examined by comparison with those obtained through the method of harmonic balance and direct numerical integration. Additionally, using the bifurcation diagrams, the effects of the state-dependent time delay and nonlinear stiffness on the chatter vibration behaviors were examined.

Nonlinear free vibrations of functionally graded nanobeams with surface effects

Nonlinear free vibration of functionally graded nanobeams is studied in this paper within the framework of Euler–Bernoulli beam model including the von Kármán geometric nonlinearity. It is assumed that material properties follow power law distributions through thickness direction. A second order nonlinear ordinary differential equation with quadratic and cubic nonlinear terms is obtained by using the free vibration modes of the corresponding linear problem. The direct numerical integration method is employed to find the natural frequencies of FG nanobeams with different boundary conditions. Numerical results demonstrate the normalized natural frequencies in different nanobeam dimensions, vibration amplitudes and volume fraction indices of FG material. Finally, the surface effects on the phase plane trajectory of fundamental mode time-dependent function have been shown.