Newmark Algorithm Research Articles

Second-Order Asymptotic Analysis and Computations of Axially and Spherically Symmetric Piezoelectric Problems for Composite Structures

The second-order coupled piezoelectric models based on the multi-scale asymptotic expansion method are developed for the axially and spherically symmetric composites. The governing piezoelectric equations are compactly formulated in cylindrical and spherical coordinates, and the composite domains are assumed to be periodically occupied by the representative cells. The multi-scale asymptotic expansions for the displacement and the electric potential are formally defined and the effective elastic, piezoelectric, and dielectric coefficients are expressed in terms of the microscopic functions defined on the cell domain. Particularly, the cell solutions and the homogenized solutions for the plane axisymmetric and spherically symmetric problem are derived analytically. The corresponding finite element procedure is proposed, in which the Newmark algorithm is applied to construct the computational scheme in the temporal domain. Numerical experiments are carried out to simulate both the static and dynamic asymptotic behavior of the space axisymmetric and the one-dimensional plane axisymmetric structures. It is validated from the numerical examples that the asymptotic models proposed in the current work are effective to capture the macroscopic performance of the piezoelectric structures and the second-order expansions of the solutions is essential for obtaining the correct distributions of the stress and electric displacement.

Fluid‐structure interaction in two‐phase flow using a discrete forcing method

SummaryThe numerical simulation of interaction between structures and two‐phase flows is a major concern for many industrial applications. Using a discrete forcing method (see the work of Benguigui et al) (implemented in a multiphase CFD code based on a two‐fluid approach) to track the solid motion in two‐phase flow, an iterative fluid‐structure coupling is developed to allow free‐motion of multiple solids (with any kind of geometry) due to two‐phase fluid forces. As the fluid‐structure interface is located thanks to a time and space dependent porosity on a cartesian grid, the fluid force computation is accommodated to the interface tracking method. A Newmark algorithm is used to estimate the solid motion. The iterative coupling is addressed in detail going from the algorithm to the determination of its convergence parameter. Three application cases are proposed to validate the method from motion under a single‐ to a two‐phase flow.

Analyses of Dynamic Behavior of Vertical Axis Wind Turbine in Transient Regime

In this paper, the dynamic behavior of a one-stage bevel gear used in vertical axis wind turbine in transient regime is investigated. Linear dynamic model is simulated by fourteen degrees of freedom. Gear excitation is induced by external and internal sources which are, respectively, the aerodynamic torque caused by the fluctuation of input wind speed in transient regime and the variation of gear mesh stiffness. In this study, the differential equations governing the system motion are solved using an implicit Newmark algorithm. In fact, there are some design parameters, which influence the performance of vertical axis wind turbine. In order to get the appropriate aerodynamic torque, the effect of each parameter is studied in this work. It was found that the rotational speed of the rotor shaft has a significant effect on the aerodynamic torque performance.

A waveform relaxation Newmark method for structural dynamics problems

In the conventional Newmark family for time integration of hyperbolic problems, both explicit and implicit methods are inherently sequential in the time domain and not well suited for parallel implementations due to unavoidable processor communication at every time step. In this work we propose a Waveform Relaxation Newmark (WRN $$_\beta $$ ) algorithm for the solution of linear second-order hyperbolic systems of ODEs in time, which retains the unconditional stability of the implicit Newmark scheme with the advantage of the lower computational cost of explicit time integration schemes. This method is unstructured in the time domain and is well suited for parallel implementation. We consider a Jacobi and Gauss–Seidel type splitting and study their convergence and stability. The performance of the WRN $$_\beta $$ algorithm is compared to a standard implicit Newmark method and the obtained results confirm the effectiveness of the Waveform Relaxation Newmark algorithm as a new class of more efficient time integrators, which is applicable, as shown in the numerical examples, to both the finite element method and meshfree methods (e.g. the reproducing kernel particle method).

Crack propagation in dynamics by embedded strong discontinuity approach: Enhanced solid versus discrete lattice model

In this work we propose and compare the two models for crack propagation in dynamics. Both models are based on embedded strong discontinuities for localized cohesive type crack description and both provide the advantage to not to require tracking algorithms. The first one is based on discrete lattice approach, where the domain is discretized with Voronoi cells held together prior to crack occurrence by cohesive links represented in terms of Timoshenko beams. The second one is based on constant strain triangular solid element. In both models, propagation of cracks activates enhancements in the displacement field leading to embedded strong discontinuities. The latter remain localized inside the element, regulated by the localized traction separation behavior defined through exponential softening law. Thus, the both models provide the result that remain mesh-independent, with fracture energy as the model parameter. We show that implementation in dynamics framework can be obtained by adding inertial effects without modifying the existing quasi-statics models. In order to provide reliable results, classical implicit Newmark algorithm can be used for time integration. The two presented models are subjected to dynamic crack propagation benchmarks, where detailed analysis on strain, kinetic, plastic free and dissipated energy during simulation is verified by comparison to the amount of total work which is introduced into the system. The main strength of the proposed approach is that cracks initiation, propagation, their coalescence, merging and branching are automatically obtained without any tracking algorithms. In addition, since the discontinuities remain localized inside elements, accurate results can be obtained even with coarser grids, leading to efficient methodology capable of capturing complex crack patterns in dynamics.

ACCURATE COMPUTATION ON DYNAMIC SIFS USING IMPROVED XFEM

Compared to the standard extended finite element method (XFEM), the improved XFEM overcomes, in theory, the linear dependence and the ill-conditioning issues, and improves in magnitude of orders the efficiency of linear system solve. In particular, in dynamic crack propagation problems, thanks to the exclusion of the extra dofs on crack tip enriched nodes, the new method eliminates the issue of energy inconsistency or the inconservitive energy transfer caused by dof dynamics, and provides more accurate dynamic stress intensity factors (DSIFs) with much less numerical oscillation. To the best of our knowledge, numerical solution on DSIFs for crack propagation under dynamic loading remains engineeringly unsatisfied. In this paper, the extra-dof free XFEM is extended to implicitly dynamic crack propagation problems-still a remaining difficulty for the current XFEMs. The implicit Newmark algorithm is used for time discretization. A dynamic interaction integral method is employed for DSIFs for both stationary and moving cracks under dynamic loading. Compared with the interaction integral method for static cracks, the dynamic method considers the effects of crack growth speed and inertia terms. The paper investigates in detail the influences of element size, mass matrix formulation, time step size, crack tip enriched zone size, inertia term, crack growth speed, and J-domain mesh/cell sizes of the interaction integral. Numerical tests show satisified accuracy and efficiency of the new method for dynamic crack problems. In particular on the challenging benchmark problem Mode I semi-infinite stationary and then moving crack, the new improved XFEM provides the best DSIFs up to the time of publication.

Nonlinear flutter analysis of composite panels

Nonlinear panel flutter is an interesting subject of fluid–structure interaction. In this paper, nonlinear flutter characteristics of curved composite panels are studied in very low supersonic flow. The composite panel with geometric nonlinearity is modeled by a nonlinear finite element method; and the responses are computed by the nonlinear Newmark algorithm. An unsteady aerodynamic solver, which contains a flux splitting scheme and dual time marching technology, is employed in calculating the unsteady pressure of the motion of the panel. Based on a half-step staggered coupled solution, the aeroelastic responses of two composite panels with different radius of R = 5 and R = 2.5 are computed and compared with each other at different dynamic pressure for Ma = 1.05. The nonlinear flutter characteristics comprising limited cycle oscillations and chaos are analyzed and discussed.

Influence of foundation mass and surface roughness on dynamic response of beam on dynamic foundation subjected to the moving load

In this paper, Improved Moving Element Method (IMEM) is used to analyze the dynamic response of Euler-Bernoulli beam structures on the dynamic foundation model subjected to the moving load. The effects of characteristic foundation model parameters such as Winkler stiffness, shear layer based on the Pasternak model, viscoelastic dashpot and characteristic parameter of mass on foundation. Beams are modeled by moving elements while the load is fixed. Based on the principle of the publicly virtual balancing and the theory of moving element method, the motion differential equation of the system is established and solved by means of the numerical integration based on the Newmark algorithm. The influence of mass on foundation and the roughness of the beam surface on the dynamic response of beam are examined in details.

Stable implicit time-integration of flexible rotating structures—explanation for instabilities and concepts for avoidance

• First explanation of well-known instability problem of Newmark time-integration for flexible rotating structures is given. • Academic solutions for instability issues (e.g. “stiffness scaling”) are presented. • Concept for avoidance of instability problems is shown (improved time-step control with detection of numerical issues). Instabilities occurring during the implicit time-integration are still handicapping a time-efficient solution of large FEM systems of equations. Especially the simulation of flexible rotating structures is barely mastered by implicit FEM codes. For this, the Newmark algorithm and related algorithms are used for many years. Here, we derive the reasons for the mentioned inevitable numerical issues and present concepts that lead to an efficient and stable solution.

Numerical Solutions for Nonlinear High Damping Rubber Bearing Isolators: Newmark’s Method with Netwon-Raphson Iteration Revisited

AbstractNumerical methods for the solution of dynamical problems in engineering go back to 1950. The most famous and widely-used time stepping algorithm was developed by Newmark in 1959. In the present study, for the first time, the Newmark algorithm is developed for the case of the trilinear hysteretic model, a model that was used to describe the shear behaviour of high damping rubber bearings. This model is calibrated against free-vibration field tests implemented on a hybrid base isolated building, namely the Solarino project in Italy, as well as against laboratory experiments. A single-degree-of-freedom system is used to describe the behaviour of a low-rise building isolated with a hybrid system comprising high damping rubber bearings and low friction sliding bearings. The behaviour of the high damping rubber bearings is simulated by the trilinear hysteretic model, while the description of the behaviour of the low friction sliding bearings is modeled by a linear Coulomb friction model. In order to prove the effectiveness of the numerical method we compare the analytically solved trilinear hysteretic model calibrated from free-vibration field tests (Solarino project) against the same model solved with the Newmark method with Netwon-Raphson iteration. Almost perfect agreement is observed between the semi-analytical solution and the fully numerical solution with Newmark’s time integration algorithm. This will allow for extension of the trilinear mechanical models to bidirectional horizontal motion, to time-varying vertical loads, to multi-degree-of-freedom-systems, as well to generalized models connected in parallel, where only numerical solutions are possible.

Dynamic response analysis of aero hydraulic pipeline system under pump fluid pressure fluctuation

The pressure fluctuation excited by the hydraulic pump can cause serious vibration in aero hydraulic pipeline system, which poses a serious threat to the safety of the aircraft. Therefore, an effective method for predicting the fluid pressure fluctuation and dynamic response of pipeline is strongly recommended. In this paper, a comprehensive model for predicting the dynamic response of pipeline, which includes the Poisson coupling, friction coupling, the vibration damping, Coriolis, and centrifugal forces is proposed. A numerical code is presented to solve the hydraulic pipeline equations. In this code, the hydraulic equations are solved by the method of characteristics and the dynamic equations of pipeline are solved by the finite element method combined with the Newmark algorithm. The numerical code is validated through the comparison of the dynamic response for a typical hydraulic pipeline with experiment. The obtained results indicate that the current combining method of characteristics and finite element method approach can predict the dynamic response of hydraulic pipeline with sufficient accuracy, which can serve as an efficient tool in the design and maintenance of aero hydraulic pipeline.

An Extended Newmark-FDTD Method for Complex Dispersive Media

Based on polarizability in the form of a complex quadratic rational function, a novel finite-difference time-domain (FDTD) approach combined with the Newmark algorithm is presented for dealing with a complex dispersive medium. In this paper, the time-stepping equation of the polarization vector is derived by applying simultaneously the Newmark algorithm to the two sides of a second-order time-domain differential equation obtained from the relation between the polarization vector and electric field intensity in the frequency domain by the inverse Fourier transform. Then, its accuracy and stability are discussed from the two aspects of theoretical analysis and numerical computation. It is observed that this method possesses the advantages of high accuracy, high stability, and a wide application scope and can thus be applied to the treatment of many complex dispersion models, including the complex conjugate pole residue model, critical point model, modified Lorentz model, and complex quadratic rational function.

Investigation on random vibration of a drillstring

This paper investigates the axial-torsional coupled vibration of a drill-string under combined deterministic and random excitations. Finite element method (FEM) is used to model the system. The random excitation at the bit-rock interaction, which is considered in the bit axial direction, is treated as Gaussian white noise. Statistic linearization is first applied to find a equivalent linear dynamic system which is then solved with stochastic Newmark algorithm. The statistics of the responses, including the means and standard deviations of the bit axial displacement and rotational velocity are obtained and analyzed.

Time-domain analysis of viscoelastic systems

The use of constrained viscoelastic materials has been regarded as a convenient strategy to reduce noise and vibrations in many types of industrial applications. The presence of local nonlinearities in the system or the implementation of a hyper-visco-elastic behaviour law, cannot be appropriately dealt with in the frequency domain and require the analysis to be performed in the time domain. This work aims at presenting a general framework for the computation of time responses of viscoelastically damped systems, by using two-step recurrence formulas involving internal variables into an unconditionally stable Newmark time discretization scheme. After validating the implementation of the methods on a sandwich beam, a numerical analysis of the modified Newmark algorithms is carried out. The methodology is then applied to a structure with more complex geometry to assess memory and CPU requirements.

Load identification in one dimensional structure based on hybrid finite element method

The new hybrid elements are proposed by combing modified Hermitian wavelet elements with ANASYS elements. Then hybrid elements are substituted into finite element formulations to solve the load identification. Transfer matrix can be constructed by using the inverse Newmark algorithm and hybrid finite element method. Loads can obtain through the responses and the transfer matrix. Load identification law was studied under different excitation cases in rod and Timoshenko beam. Regularization method is adopted to solve ill-posed inverse problem of load identification. Compared with ANSYS results, hybrid elements and HCSWI elements can accurately identify the applied load. Numerical results show that the algorithm of hybrid elements is effective. The accuracy of hybrid elements and HCSWI elements can be verified by comparing the load identification result of ANASYS elements with the experiment data. Hermitian wavelet finite element methods have high accuracy advantage but it is difficult to apply the engineering practice. In practical engineering, complex structure can be analyzed by using the hybrid finite element methods which can be obtained the high accuracy in the crucial component.

Forced vibration analysis of functionally graded beams by the meshfree boundary-domain integral equation method

Forced vibration of two-dimensional functionally graded beams is studied in this paper by a developed meshfree boundary-domain integral equation method. Material properties of the functionally graded beams are assumed varying continuously either in longitudinal or transvers direction following the exponential function. The boundary-domain integral equations are derived by using the elastostatic fundamental solutions based on the two-dimensional elastic theory. Radial integral method (RIM) is employed to transform the domain integrals into boundary integrals. A meshfree scheme is achieved through assuming the displacements and accelerations in the domain integrals by a combination of the radial basis function and polynomials with time dependent coefficients. Wilson-θ, Houbolt as well as two kinds of damped Newmark's algorithms are applied to accomplish the time integration. The forced vibration of the functionally graded beam subjected by the harmonic loading and transient loading are investigated in detail. Numerical examples demonstrate that the four mentioned time integral schemes are all adapted well to the developed meshfree boundary-domain integral equation method for analyzing the forced vibration of homogeneous structures. For the analysis of FG structures, it is shown that the damped Newmark's algorithm can achieve more stable and accurate results.

A modified layered-section method for responses of fire-damaged reinforced concrete beams under static and blast loads

Reinforced concrete structures are currently under the threat of both fire and blast. The absence of theoretical methods demonstrates a drawback in the assessment of blast-resistant structures after exposure to fire. A modified layered-section method was developed in this article, which was not only able to determine the complete static resistance–deflection curves of fire-damaged reinforced concrete beams but also able to predict the responses of reinforced concrete beams subjected to blast after fire exposure. The high-temperature effects and the strain-rate effects were included in the concrete and steel material models in the proposed method. A corresponding calculation program FBBA was also compiled based on the explicit Newmark algorithm on the platform of Maple software. The developed method and program were validated by the existing test results. Analytical results showed that after fire exposure, the reinforced concrete beams show significant degradation in the residual bearing capacity, but increase in the ductility. The higher the steel reinforcement ratio, the more degradation the bearing capacity of reinforced concrete beams after fire exposure suffers. The blast resistance of the fired reinforced concrete beams was underestimated without considering the strain-rate effects or just considering the average strain-rate effects.

Effects of variable loading conditions on the dynamic behaviour of planetary gear with power recirculation

Abstract Variable loads to which gearboxes are subjected are considered as one of the main sources of non-stationarity in these transmissions. In order to characterise their dynamic behaviour in such conditions, a torsional lumped parameter model of a planetary gear with power recirculation was developed. The model included time varying loading conditions and took into account the non-linearity of contact between teeth. The meshing stiffness functions were modelled using Finite Element Method and Hertzian contact theory in these conditions. Series of numerical simulations was conducted in stationary conditions, with different loading conditions. Equation of motion was solved using Newmark algorithm. Numerical results agreed with experimental results obtained from a planetary gear test bench. This test bench is composed of two similar planetary gears called test planetary gear set and reaction planetary gear set which are mounted back-to-back so that the power recirculates through the transmission. The external load was applied through an arm attached to the free reaction ring. Data Acquisition System acquired signals from accelerometers mounted on the rings and tachometer which measured instantaneous angular velocity of the carrier’s shaft. The signal processing was achieved using LMS Test.Lab software. Modulation sidebands were obtained from the ring acceleration measurements as well as a non-linear behaviour in case of variable loading resulted by a transfer of the spectral density from the fundamental mesh stiffness to its second harmonic.

Modeling and simulating of V-shaped piezoelectric micro-cantilevers using MCS theory considering the various surface geometries

Atomic force microscopy (AFM) is widely used as a tool in studying surfaces and mechanical properties of materials at nanoscale. This paper deals with mechanical and vibration analysis of AFM vibration in the non-contact and tapping modes for V-shaped piezoelectric micro-cantilever (MC) with geometric discontinuities and cross section variation in the air ambient. In the vibration analysis, Euler-Bernoulli beam theory based on modified couple stress (MCS) theory has been used. The governing equation of motion has been derived by using Hamilton's principle. By adopting finite element method (FEM), the MC differential equation has been solved. Damping matrix was considered in the modal space. Frequency response was obtained by using Laplace transform, and it has been compared with experimental results. Newmark algorithm has been used based on constant average acceleration to analyze time response of MC, and then time response results in the vibration mode, far from the sample surface have been compared with experimental data. In vicinity of sample surface, MC is influenced by various nonlinear forces between the probe tip and sample surface, including van der Waals, contact, and capillary forces. Time response was examined at different distances between MC base and sample surface, and the best distance was selected for topography. Topography results of different types of roughness showed that piezoelectric MC has been improved in the air ambient. Topography showed more accurate forms of roughness, when MC passes through sample surface at higher frequencies. The surface topography investigation for tapping and non-contact modes showed that using of these two modes are suitable for topography.

Benefits of metamodel-reduction for nonlinear dynamic response analysis of damaged composite structures

In this paper, a novel method for damage prediction and dynamic behaviour analysis of laminate composite structures is proposed and investigated. The dynamic behaviour of transversely isotropic layers is expressed through elasticity coupled with damage using a phenomenological macro-model for cracked composite structures made of polymer reinforced with long glass fibres. The damage is fully described by a single scalar variable whose evolution law is expressed through the maximum dissipation principle. Using the classical linear Kirchhoff–Love theory of plates and considering the damage-induced nonlinearity, the obtained nonlinear dynamic equations are solved in time domain using a Newmark algorithm. To reduce the computational costs, a metamodel-reduction for damage localization and quantification is proposed where the Artificial Neural Networks (ANNs) and Craig–Bampton reduction methods are combined. Extracted stresses from finite element analysis are used as input for a feed-forward ANNs to estimate the damage severity. The Craig–Bampton reduction method is introduced as a Component Mode Synthesis (CMS) to investigate the case of assembled structures locally damaged. Numerical simulations show that the damage modifies significantly the dynamic properties restricted to the eigenfrequencies reduction. The designed feed-forward ANNs was verified and it provides promising results regarding severity and location of the damage. Moreover, the trained ANNs provide a quick response for damage level prediction in online procedure which permits to significantly reduce the computational costs.