Newmark Method Research Articles

An effective model for bolted flange joints and its application in vibrations of bolted flange joint multiple-plate structures: Theory with experiment verification

Bolted flange joints are widely used in various engineering structures to combine two or more substructures together. However, the understanding of mechanical mechanism of bolted flange joints is still insufficient due to the lack of efficient mathematical models. This paper proposes an effective mathematical model for bolted flange joints, which is then employed to conduct vibration analysis on bolted flange joint multiple-plate structures. During the modeling process, the flange is modeled by the Kirchhoff plate theory, and the bolted joints are modeled by the two-dimensional face constraint in bolted joint affected region (BJAR). The artificial spring technique is adopted to simulate the face constraint. The friction contact model and equivalent linearization technique are adopted to analyze the stick and slip states of bolted joints. By adopting the Chebyshev orthogonal polynomials as admissible functions, governing equations are derived. Then, the Rayleigh-Ritz method is applied to study the free vibration characteristics, and the Newmark-beta method is employed to study the vibration response under base excitation. The experimental studies are performed on the bolted flange joint two-plate structure (BFJTPS) to illustrate the effectiveness of the proposed theoretical model. The results indicate that the proposed mathematical model can well predict the vibration characteristics of multiple-plate structures under both bolt looseness and no-looseness conditions. The soft nonlinearity phenomenon under bolt looseness condition can be successfully revealed by the model.

Numerical Solution of Hydrodynamic Efficiency Equations for an Oscillating Water Column Wave Energy Converter Using the Method of Fundamental Solutions

The oscillating water column (OWC) converter, a prominent and widely adopted device for capturing wave energy, has yet to attain commercialization due to its intricate hydrodynamic behavior. Consequently, numerous laboratory and numerical studies have been carried out on OWC converters. One of the significant parameters in the study and comparison of different converters is their hydrodynamic efficiency parameter. Hence, the primary objective of this study is to examine the efficiency of an oscillating water column converter using the meshless method of fundamental solutions (MFS) to solve the governing equations. To address the challenges posed by the low wave steepness and to optimize computational efficiency, a complete linearization of two nonlinear sources at the boundary conditions inside and outside the OWC chamber was undertaken. Next, the boundary conditions were temporally discretized using three distinct methods: the Newmark, the forward differentiation formula, and the Runge-Kutta method of the fourth order. Subsequently, a comparison was made among these methods, revealing that the Newmark method exhibited a higher convergence rate to the solution. Using the developed numerical model, a more precise examination of the converter's efficiency, under various sloped bottom geometries, was conducted, surpassing the level of accuracy achieved in prior studies. Two key parameters, namely the opening, and slope, were identified as critical factors influencing the efficiency of converters considering sloped bottoms. The results indicated that the efficiency of the converter decreases with an increase in the slope and a decrease in the opening. Furthermore, to effectively interpret the efficiency of the converter, a suitable parameter was introduced the disparity in the free surface elevation inside and outside the chamber. This parameter proved valuable in providing insights into the converter's performance.

A refined first-order shear deformation theory for large dynamic and static deflection investigations of abruptly/harmonically pressurized/blasted incompressible circular hyperelastic plates

In this paper, nonlinear large-deformation dynamic and static responses of the suddenly and harmonically pressurized or blasted compliant incompressible hyperelastic circular plates are investigated through the development of a novel enhanced circular plate theory that in contrast to the available plate theories, utilizes transverse normal strains whose order is much higher than those of the in-plane strains. For this reason, no shear correction factor is needed for the new plate theory. The deformation gradient and Green's strain tensors are first determined based on the first-order shear deformation plate theory as predictors and then, corrected by simultaneously including the transverse compliance and volume conservation concepts in the neo-Hookean framework of hyperelasticity. The equations of motion are obtained using Hamilton's principle. The resulting highly nonlinear equations that include both large deformation and constitutive nonlinearities are solved by a hybrid iterative/updating technique that utilizes a combination of the differential quadrature (DQ) spatial-discretization and Newmark's time-marching methods, after an exhaustive order analysis for the identification of the extremely small terms. The effects of the changes in geometric parameters, material properties, and loading rate and type are investigated on the dynamic lateral deflection of the plate and the approximate fundamental natural frequency. Results reveal that the contribution of the higher vibration modes in the responses is much more remarkable in the high-rate and high-frequency loadings. Furthermore, the deformed shape of the plate that indicates the superimposed effects of the higher vibration modes is plotted for successive time instants.

Recommendation of new design spectra for Iran using modified Newmark method

The Newmark design spectra are commonly adopted in seismic codes to calculate design spectra, while these spectra generally differ from the statistically driven ones. This study aims to re-construct the Iranian design spectra by implementing a modified Newmark method on an extensive database of previous earthquakes in Iran. Three sets of recorded earthquakes are considered to evaluate the effects of parameters such as source-to-site distance, magnitude of ground motion, and shear-wave velocity. Considering that the selected earthquakes have different frequency contents, the suitable frequency ranges for each earthquake category are obtained. Subsequently, a statistical approach is adopted to calculate amplification factors and spectral bounds of the site design spectrum of Iran for three different site categories as a function of ground motion's magnitude. The calculated coefficients can be used to modify the Newmark spectral values in displacement, velocity, and acceleration-sensitive regions to obtain suitable design spectra. The results indicate that, in general, the proposed design spectra can provide a better representative of the earthquakes in Iran compared to the current Iranian seismic design code.

Transient state analysis of a rub-impact rotor system during maneuvering flight

Maneuvering flight substantially affects the dynamic behavior of rotors; particularly, such flight may cause rubbing between a rotor and stator, which is one of the most serious damages in aircraft engines. In this paper, a nonlinear dynamic model for describing the dynamic characteristics of a rub-impact rotor system during maneuvering flight is established based on the Lagrange equations. Subsequently, numerical simulations employing the Newmark method are performed, delving into the detailed discussion of the influence of parameters such as rotational speed and maneuvering flight on the transient and steady-state responses of the rotor system. The effect mechanism of maneuver load and its coupling with rub impact is revealed. The results show that the impact response induced by maneuvering flight is more obvious in the subcritical state than in the supercritical state. The additional stiffness and damping are also induced; in particular, the additional damping has a coupling effect. Moreover, the rub impact imposes an additional constraint on the rotor system, thereby weakening the influence of the maneuver load and becoming the major factor that determines the dynamic behavior of the rotor system at high speeds.

An edge-based smoothed finite element method combined with Newmark method for time domain acoustic wave scattering problem

Time domain obstacle scattering widely exists in the scientific fields of geophysics, remote sensing, seismology, and medicine. However, it is difficult to solve the time domain scattering problems with arbitrarily shaped obstacles only within the fixed domain of interest. In this paper, an edge-based smoothed finite element method using linear triangular elements (ES-FEM-T3) combined with the Newmark method is proposed for the two-dimensional time domain acoustic scattering by an arbitrarily shaped obstacle. In the algorithm, a time domain transparent boundary condition (TBC) and the Newmark method are introduced to transform the time domain problem in an unbounded domain into a series of steady-state PDEs on a bounded domain, which is suitable for an arbitrary shaped obstacle scattering problem. In the numerical discretization, ES-FEM-T3 is used to construct a close-to-exact stiffness matrix by the smoothed gradient technique, making it space stable and obtaining more accurate results compared to FEM-T3. Especially, through in advance calculating the function σ(t), the convolution term in the time-domain TBC can be quickly calculated. Finally, through the numerical experiments, the effectiveness of the proposed method is demonstrated. Especially, ES-FEM-T3 can better improve the time convergence stability of the Newmark method and have better accuracy at γ = 0.5 compared to FEM-T3.

Structures with varying discontinuities and curvatures: A dynamic analysis approach by the p-version finite element method

The present paper investigates the geometrically nonlinear time domain dynamic analysis of a coupled Timoshenko beam-beam or beam-arch mechanical system. The coupled structure is modelled with a variable discontinuity in an elastic layer, which represents a real case from technical practice where there is no continuous distribution of the elastic layer or the stiffness of the layer is changed by other influences at different locations. A modified p-version finite element method is developed for the vibrations of a shear deformable coupled beam system with a discontinuity in an elastic layer. The main contribution of this work is the discovery of coupled effects and phenomena in the simultaneous vibration analysis of varying discontinuity and varying curvature of the newly modelled coupled mechanical system. New general mode shapes are presented, and the forced vibrations in the time domain are analyzed using the Newmark method. The study compares the results for various locations and types (size) of discontinuity in an elastic coupling layer and various curvatures of the lower supported beam. It explains the cases in which the influence of curvature takes precedence over discontinuity and vice versa in achieving a steady-state regime of vibrations. The analysis results are valuable and have broader applications in the field of solids and structures.

Crack phase‐field enhanced finite cover method for dynamic fracture at finite strain

SummaryThis study presents an enhancement of the diffusive‐discrete crack transition scheme (10.1002/nme.7169) to describe dynamic fracture at finite strain. In the enhanced scheme, the crack initiation, propagation, and bifurcation processes are determined from an energy minimization problem based on crack phase‐field theory, and the predicted diffusive crack is replaced by the discrete representation using the finite cover method. In the meantime, numerical damping is introduced to maintain computational stability and avoid distortion of the physical mesh in the finite cover context. By taking advantage of the features of the diffusive‐discrete crack transition scheme, the proposed approach enables us to stably simulate a series of dynamic fracture events involving crack initiation at an arbitrary location, propagation, and bifurcation in arbitrary directions, arbitrary divisions of an original object into multiple portions, and independent motions of divided portions. After spatial and temporal discretizations by the finite cover method and the Newmark method are described, as well as the simulation algorithm of the enhanced finite cover‐based staggered iterative procedure for dynamic fracture, several representative numerical examples are presented to demonstrate the performance and capabilities of the developed approach.

Rocking sensitivity of a dual‐block stack ‐ Numerical simulation and experimental evidence

AbstractHistoric monuments, drywall structures, and graphite blocks in AGR nuclear power plants are block‐like structures that have to withstand rocking when subject to seismic excitation of their base, which can lead to overturning of some of their components and results in the collapse of the whole structure. We revisit the known nonlinear equations of motion for a dual‐block stack and present the conditions for transition between the eight possible rocking configurations (due to initiation of rocking, opening of new contacts, and collisions between blocks). An algorithm for the numerical simulation of rocking of the dual‐block stack is developed using the Newmark integration method, the Newton‐Raphson iteration method, and a novel contact detection and resolution procedure. The algorithm is used to evaluate rocking stability of five dual‐block stacks, one of which is compared to the results available in the literature. In parallel, a novel experimental program is designed and implemented, to validate the numerically obtained results using a shaking table. While most of the excitation conditions leading to stable rocking and limit values leading to overturning have been successfully validated, some discrepancies between the numerically and experimentally obtained results still exist and point to the need for improvement of the algorithm used, possibly through a more realistic energy‐loss mechanism. Most importantly, we have confirmed the known theoretical prediction that splitting a single block into two half‐size blocks benefits its rocking stability.

Large-eddy simulation of cultivated fields with Cyperus malaccensis Lam

ABSTRACT In this study, the honami phenomenon in Cyperus malaccensis Lam. was reproduced to establish a simulation technique that can accurately reproduce the waves of it and predict the wind flow over it and through it, which is related to its growth. The analysis code was constructed on the basis of a large-eddy simulation (LES). The LES model is based on the nonrotating coherent-structure Smagorinsky model (NRCSM), which is an expansion of the Smagorinsky model. In addition, this code is solved for a bundle of reaped rice plants, which are based on the vegetation movement, using the Newmark-beta method with an LES to analyze the movement of vegetation and air in the field. As a result, vortex structures and vegetation oscillations were visualized. We observed not only the streamwise vortex structures near the tips of C. malaccensis Lam. but also characteristic structures that sink into the vegetation simultaneously with vegetation shaking. Consequently, we found that the vibration of C. malaccensis Lam. is influenced by air inflow within the planting.

Dynamic finite element modeling of axially functionally graded Timoshenko microbeams under a moving mass

This paper presents a finite element model for dynamic analysis of axial functionally graded (AFG) microbeams subjected to amovingmass. Thematerial properties of themicrobeams are considered to vary in the axial direction by a power-law function, and they are evaluated byMori-Tanaka scheme. The first-order shear deformation theory is employed in conjunction with the modified couple stress theory to establish the differential equations of motion for the AFG microbeams. A two-node beam element with six degrees of freedom was derived and used in combination with Newmark method to solve the equations of motion and to compute the dynamic response of the microbeams. Numerical investigations are carried out on AFG microbeam with simply supported ends. The proposed method is validated by comparing with results in previous work. The effects of dimension-less material length scale parameters, the material distribution and the moving mass parameters on the dynamic characteristics of the AFG microbeams are studied and discussed in detail.

Hygrothermally-Induced Vibration Analysis of Porous FGM Rectangular Mindlin Plates Resting on Elastic Foundation

The nonlinear vibration response of rectangular plates made of functionally graded porous materials (FGPs) induced by hygrothermal loading is investigated in this article using a numerical approach. The effect of elastic foundation on the vibrations is taken into account according to the Winkler–Pasternak model. Hygroscopic stresses produced due to nonlinear rise in moisture concentration are also considered. The temperature-dependent material properties of plate are computed based on the modified Voigt’s rule of mixture and Touloukian experiments for even and uneven distribution patterns of porosity. Within the framework of the first-order shear deformation plate theory and von-Kármán nonlinearity, Hamilton’s principle is utilized in order to derive the equations of motions. To achieve the temporal evolution of maximum lateral deflection of hygrothermally-induced plates, the generalized differential quadrature (GDQ) and Newmark integration methods are employed. Selected numerical results are presented to study the influences of temperature distribution, porosity volume fraction, moisture concentration, geometrical parameters, elastic foundation parameters and FG index on the geometrically nonlinear vibrations of FG porous plates with various boundary conditions.

Dynamical analysis of a time-varying length rope-driven system with boundary displacement excitation based on time-varying length finite element method

This paper developed a finite element method for the time-varying length rope-driven system under the Eulerian description since the transient responses of the time-varying length rope-driven system induced by boundary displacement excitation cannot be considered accurately in the prior work. The displacement excitation can be considered directly by substituting the boundary conditions after preprocessing that matches the subsequent time discrete method—Newmark-Beta method in this paper—into the numerical calculation. Meanwhile, the treatment of the mixed boundary conditions with a concentrated mass is introduced under the Eulerian description. The proposed method is validated by a basic model with an analytical solution and compared with the Galerkin method and by a sophisticated model derived from an actual elevator from the perspectives of complex mode analysis and linear-nonlinear time domain calculation, respectively. The results show that the proposed method can effectively deal with complex boundary conditions and accurately capture the transient wave response caused by displacement boundary excitation. Compared with the Galerkin method, whose accuracy depends on the forms and orders of the selected trial functions, the proposed method has better adaptability and accuracy in dynamical analysis of a time-varying length rope-driven system.

Dynamic analysis of 2DFGM porous nanobeams under moving load with surface stress and microstructure effects using Ritz method

This paper investigates the dynamic behavior of micro/nanobeams made of two-dimensional functionally graded porous material (2DFGPM) under accelerated, decelerated, and uniform moving harmonic load, using surface elasticity and modified couple stress theories. The key feature of this formulation is that it deals with a higher order shear deformation beam theory. The non-classical equilibrium equations are developed using Lagrange's equation and the concept of physical neutral surface. The equations of motion are derived using the same approach, accounting for the porosity effect and the modified power-law distribution of material properties. The trigonometric Ritz method is used with sinusoidal trial functions for the displacement field, and the Newmark method is applied to obtain the dynamical response of 2DFGPM nanobeams. The results are compared with previous studies, and the impact of critical parameters such as gradation indices, volume fraction ratio, pattern of porosity, velocity, frequency, and motion type of the applied force are explored. This study highlights the importance of considering the porosity effect, as neglecting it can lead to significant errors in the predicted results. Additionally, the study found that the accelerated and decelerated motions of the applied load have a greater impact on the dynamical deflection of 2DFGPM nanobeams than the uniform motion. The findings of this study can provide guidance for the optimal design of micro/nanobeams subjected to a moving force with multifunctional properties.

A Size-Dependent Viscoelastic Model for Microbars with Variable Cross-Section

This paper presents a model for microbars with variable cross-sections using the Kelvin–Voigt model for viscoelastic material, accounting for size-dependent effects based on strain gradient theory. The size-dependent dynamic equations for the rod, which consider the variable cross-sectional area, are obtained through the extended Hamilton’s principle. These equations are then reduced in order using the Galerkin method and solved in the steady state using the harmonic response form and the algebra of complex numbers. To solve the equations from the transient state to the steady state, a combined method is implemented using the Grünwald–Letnikov derivative technique and the Newmark method. Furthermore, a model and analysis based on the finite element method are presented to validate the results. In the results section, various factors such as size-dependent effects, the order of the fractional derivative, the amount of the viscoelastic coefficient, and the shape of the section area are examined through the time history graph, frequency response, and maximum displacement in terms of force. The results demonstrate that the transient response converges to the stable response after a certain period of time. Moreover, it is observed that decreasing the order of the fractional derivative in the pre-resonance range leads to a decrease in response sensitivity, while in the resonance frequency range, the sensitivity increases with the increase in order.

Hygro-thermo-mechanical coupling behavior of porous FG-GNPRC annular plates considering aggregation of GNPs

The research on graphene nanolayers has important theoretical and practical significance for promoting the application of advanced composite materials in the field of engineering. This paper investigates the mechanical properties of FG-GNPRC annular plates with voids in hot and humid environments. Considering the aggregation effect and aggregation type of graphene nanoplatelets (GNPs) in GNP reinforced composites (GRCs), the random distribution pattern of GNPs in GRCs, and the gradient distribution of GNPs in functional gradient GNPRC (FG-GNPRC), the mechanical model of FG-GNPRC annular plate with variable thickness under hygro-thermo-mechanical environment is established. Applying the Newmark method to solve the problem in the time domain, the basic equations of the coupled thermal and humidity fields as well as radial displacement are derived and solved. The effects of the randomly arranged distribution pattern of GNPs, the aspect ratio of GNPs, the aggregation types and aggregation parameters of GNPs within the clusters, the gradient index, and the variation rule of thickness on the hygro-thermo-mechanical response of the porous FG-GRC annular plates are investigated by numerical examples.

Thermally nonlinear analysis of propagating cracks under generalized thermal shock

In this paper, the thermally nonlinear response of stationary and moving cracks in isotropic media subjected to a hyperbolic thermal shock in the framework of the eXtended Finite Element Method (XFEM) is examined. For this purpose, the thermally nonlinear fully coupled Lord-Shulman (LS) model is considered as governing equations. In addition, two cases of material properties, temperature-dependent (TD) and temperature-independent (TID), are assumed. The geometry, including a predefined crack, is modeled using the XFEM for spatial discretization. For modeling the dynamic crack growth within the XFEM, the Maximum Tangential Stress (MTS) criterion is modified based on the stress field near the tip of a growing crack to calculate the dynamic crack growth angle. Besides, the nonlinear form of the Newmark integration scheme is implemented for the numerical integration in the time domain. Finally, a method based on one of the Barnet-Lothe tensors for isotropic materials is employed to extract the stress intensity factors (SIFs). In the provided examples, the effects of thermal nonlinearity, the temperature-dependency of material properties, and the LS relaxation time on the crack propagation path are explored.

The dynamic analysis of axisymmetric bodies with damping effects using the modified radial integration boundary elements method (MRIBEM)

This paper investigates the elastodynamic analysis of axisymmetric bodies under axisymmetric dynamic loads using the modified radial integration boundary elements method (MRIBEM). Purposely, the governing integral equations have been developed in cylindrical coordinates by employing the three-dimensional (3D) static fundamental solution. The axisymmetric bodies can be modeled as a two-dimensional (2D) problem, which is much simpler than 3D analysis and only requires a one-dimensional boundary mesh. In the proposed formulation, the effects of inertial and damping forces appear as 2D domain integrals. The domain variables are approximated using global radial basis functions (RBFs), and the modified radial integration method (MRIM) is employed to deal with the domain integral as the main idea of the present research. Various sample problems are provided in order to illustrate the capability of the presented method for analyzing axisymmetric bodies with concave geometry and damping effect under different types of transient loads. The Newmark and Houbolt methods were applied for time marching, and their efficiency was examined. The results of the present study were also validated by the axisymmetric FEM formulation and other available references.

Analytical and mesh-free approaches to dynamic analysis and active control of smart FGP-GPLRC beam

This paper aims to perform the dynamic analysis and active control of a functionally graded porous beam reinforced with graphene platelets (GPLs) and embedded in piezoelectric layers (smart FGP-GPLRC beam). Two types of porosity distribution combined with three dispersions of GPL pattern for the FGP-GPLRC beam are considered. The Halpin–Tsai model is used for estimating the effective elastic modulus of the FGP-GPLRC beam. The linear electric potential field through the thickness of each piezoelectric layer is assumed. Energy principles are applied to establish the equations of motion, which are then solved by two approaches. The first one is an analytical method with the Navier technique and the second one is a mesh-free method based on the polynomial basis in conjunction with the C1 Hermite interpolation technique. Newmark's integration algorithm with constant average acceleration is employed to determine the time-dependent response. The velocity feedback control algorithm is utilized to actively control the vibration response of the smart FGP-GPLRC beam. The convergence of mesh-free analysis is performed through a numerical test. Various comparative examples are carried out rigorously to verify the accuracy of the current study. Influences of porosity and GPL parameters, open- and closed-circuit states on the natural frequency, and forced vibration of the smart beam are investigated via numerical examples. In addition, the study also examines the effects of velocity feedback control gain on vibration suppression and the prevention of the beat and resonance problems occurring in the smart beam. Finally, the solution of using a collocated piezoelectric sensor/actuator pair on the same side of the FGP-GPLRC beam to guarantee the stability of the vibration control with the stretching-bending coupling effect is illustrated.

Dynamic Analysis of a Bolted Joint Rotor-Bearing System with a Blade–Casing Rubbing Fault

Bolted joints are widely used in aeroengine rotor systems to connect multiple components into an integrated structure and provide sufficient stiffness. The mechanical properties of a bolted joint have a significant effect on rotor dynamics. For modern aeroengine designs, the blade-tip clearance is gradually reduced to improve efficiency, which may lead to rubbing damage and affect safe operation. The mechanical properties of a bolted joint change significantly during the blade–casing rubbing process and influence the dynamic properties of the rotor system. Based on the finite element (FE) modeling method, a 15-node bolted joint rotor system model is established in this paper, in which the bolted joint is represented by a 2-node joint element, and the blade–casing rubbing force is considered. The Newmark method is used to solve the motion equations. The dynamic model is validated by comparing the frequency response characteristics for different numbers of blades with the results provided in other published studies. Based on the established model, the effects of the rotational speed, number of blades, and rubbing stiffness on the dynamic responses, normal rubbing forces, and bending stiffness of the bolted joint are evaluated by numerical simulation. The results show that the response amplitude and bending stiffness of the bolted joint change significantly under blade–casing rubbing faults, and the mean value of the vibration response deviates significantly from 0 as the number of blades increases. Meanwhile, the amplitude of the frequency component fVC and the maximum value of the normal rubbing force also increase as the number of blades increases. The main contribution of this paper is the establishment of a new model for a bolted joint rotor system, considering the time-varying bending stiffness of the bolted joint and the blade–casing rub fault, comparing the simulation results to obtain some general results bridging the current research gap. Meanwhile, the numerical results in this paper can provide a cognitive basis for the blade–casing rubbing fault mechanism of a bolted joint rotor system under the influence of speed, number of blades, and rubbing stiffness. The nonlinear dynamic characteristics observed in the present paper can be applied to the blade–casing rubbing fault diagnosis of turbomachines.