Newmark Time Integration Method Research Articles

Design sensitivity analysis of dynamic responses for a BLDC motor with mechanical and electromagnetic interactions

This paper presents a design sensitivity analysis of dynamic responses of a BLDC motor with mechanical and electromagnetic interactions. Based on the equations of motion which consider mechanical and electromagnetic interactions of the motor, the sensitivity equations for the dynamic responses were derived by applying the direct differential method. From the sensitivity equation along with the equations of motion, the time responses for the sensitivity analysis were obtained by using the Newmark time integration method. The sensitivities of the motor performances such as the electromagnetic torque, rotating speed, and vibration level were analyzed for the six design parameters of rotor mass, shaft/bearing stiffness, rotor eccentricity, winding resistance, coil turn number, and residual magnetic flux density. Furthermore, to achieve a higher torque, higher speed, and lower vibration level, a new BLDC motor was designed by applying the multi-objective function method. It was found that all three performances are sensitive to the design parameters in the order of the coil turn number, magnetic flux density, rotor mass, winding resistance, rotor eccentricity, and stiffness. It was also found that the torque and vibration level are more sensitive to the parameters than the rotating speed. Finally, by applying the sensitivity analysis results, a new optimized design of the motor resulted in better performances. The newly designed motor showed an improved torque, rotating speed, and vibration level.

Moving Kriging Interpolation‐based Meshfree Method for Dynamic Analysis of Structures

AbstractA novel meshfree model based on the standard element‐free Galerkin method incorporated moving Kriging interpolation (MK) is developed for free and forced vibration analysis of 2D structures. Instead of employing moving least square approximation (MLS), shape functions here are constructed by the MK method. Due to the satisfaction of the Kronecker delta function, the essential boundary conditions are thus imposed directly as the finite element method and no special techniques are required. Elastodynamic equations are transformed into a standard weak formulation and then discretized into a meshfree time‐dependent equation solved by the standard Newmark time integration method. Some numerical examples of stuctural problems in 2D are attempted, and it is found that the method is adequately accurate and stable for dynamic problems. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Dynamic analysis for a planetary gear with time-varying pressure angles and contact ratios

In this study, the dynamic responses of a planetary gear are analyzed when component gears have time-varying pressure angles and contact ratios caused by bearing deformations. For this purpose, this study proposes a new dynamic model of the planetary gear, in which the pressure angles and contact ratios change with time. The main difference from previous studies is that the present study regards the pressure angles and contact ratios as time-varying variables, while previous studies regarded them as constants. After nonlinear equations of motion for the planetary gear are derived, the dynamic responses are computed by applying the Newmark time integration method. The time responses for the present and previous studies are compared to show the effects of the time-varying pressure angles and contact ratios on the dynamic behaviors of a planetary gear. In addition, the effects of bearing stiffness on the pressure angles and contact ratios are also analyzed.

유체가 이송하는 테더가 있는 인공위성의 동특성 분석

The purpose of this study is to analyze nonlinear dynamics of a tethered satellite. The coupled non-linear equations of motion are derived by using the extended Hamilton's principle with the polar coordinate system. In order to analyze the response of tethered satellite, time responses are computed by the Newmark's time integration method. We also investigate the dynamic behavior of the system and the effects of length of tether, tip mass and conveyed fluid through the tether with time variation.

Improved 2D model of a ball bearing for the simulation of vibrations due to faults during run-up

This paper present an improved 2D bearing model for investigation of the vibrations of a ball-bearing during run-up. The presented numerical model assumes deformable outer race, which is modelled with finite elements, centrifugal load effects and radial clearance. The contact force for the balls is described by a nonlinear Hertzian contact deformation. Various surface defects due to local deformations are introduced into the developed model. The detailed geometry of the local defects is modelled as an impressed ellipsoid on the races and as a flattened sphere for the rolling balls. The obtained equations of motion were solved numerically with a modified Newmark time-integration method for the increasing rotational frequency of the shaft. The simulated vibrational response of the bearing with different local faults was used to test the suitability of the continuous wavelet transformation for the bearing fault identification and classification.

Active Control of Geometrically Non-linear Transient Response of Smart Laminated Composite Plate Integrated with AFC Actuator and PVDF Sensor

A coupled electromechanical finite element formulation for active control of geometrically non-linear transient response of laminated composite plate is studied. First-order shear deformation theory and von Karman type non-linear strain displacements are used. The plate is discritized using eight-noded quadratic isoparametric elements with five mechanical degrees of freedom and one electrical degree of freedom per node. Newton-Raphson iterative method in association with Newmark time integration method is used to solve the non-linear finite element equilibrium equation. Negative velocity feedback control algorithm is used to control the dynamic response of the smart laminated composite plate. AFC layer poled in fiber direction acting as distributed actuator and PVDF layer poled in thickness direction acting as sensor are considered. This study involves two types of actuator sensor arrangements: (1) the substrate is sandwiched between AFC actuator and PVDF sensor, called non-collocated arrangement and (2) AFC actuator and PVDF sensor are placed on top of the substrate, called collocated arrangement. The effect of piezoelectric fiber orientation in actuator layer on vibration control for both cross-ply and angle-ply laminates are examined.

Three dimensional static and dynamic analysis of thick functionally graded plates by the meshless local Petrov–Galerkin (MLPG) method

In this paper, three dimensional (3D) static and dynamic analysis of thick functionally graded plates based on the Meshless Local Petrov–Galerkin (MLPG) is presented. Using the kinematics of a three-dimensional continuum, the local weak form of the equilibrium equations is derived. A weak formulation for the set of governing equations is transformed into local integral equations on local sub-domains using a Heaviside step function as test function. In this case, governing equations corresponding to the stiffness matrix do not involve any domain integration or singular integrals. Nodal points are distributed in the 3D analyzed domain and each node is surrounded by a cubic sub-domain to which a local integral equation is applied. The meshless approximation based on the three-dimensional Moving Least-Square (MLS) is employed as shape function to approximate the field variable of scattered nodes in the problem domain. The Newmark time integration method is used to solve the system of coupled second-order ODEs. Effective material properties of the plate, made of two isotropic constituents with volume fractions varying only in the thickness direction, are computed using the Mori–Tanaka homogenization technique. Numerical examples for solving the static and dynamic response of elastic thick functionally graded plates are demonstrated. As a result, the distributions of the deflection and stresses through the plate thickness are presented for different material gradients and boundary conditions. The effects of the volume fractions of the constituents on the centroidal deflection are also investigated. The numerical efficiency of the proposed meshless method is illustrated by the comparison of results obtained from previous literatures.

테더가 있는 인공위성의 비선형 동역학 해석

The purpose of this study is to analyze nonlinear dynamics of a tethered satellite. The nonlinear equations of motion are derived by using Lagrange's equations with the polar coordinate system. In order to analyze the response of tethered satellite, time responses are computed by the Newmark's time integration method. This paper claims that the dynamic behavior of the system is changed by the effect of length of tether, mass ratio of satellites.

Improved model of a ball bearing for the simulation of vibration signals due to faults during run-up

In this paper an improved bearing model is developed in order to investigate the vibrations of a ball bearing during run-up. The numerical bearing model was developed with the assumptions that the inner race has only 2 DOF and that the outer race is deformable in the radial direction, and is modelled with finite elements. The centrifugal load effect and the radial clearance are taken into account. The contact force for the balls is described by a nonlinear Hertzian contact deformation. Various surface defects due to local deformations are introduced into the developed model. The detailed geometry of the local defects is modelled as an impressed ellipsoid on the races and as a flattened sphere for the rolling balls. With the developed bearing model the transmission path of the bearing housing can be taken into account, since the outer ring can be coupled with the FE model of the housing. The obtained equations of motion were solved numerically with a modified Newmark time-integration method for the increasing rotational frequency of the shaft. The simulated vibrational response of the bearing with different local faults was used to test the suitability of the envelope analysis technique and the continuous wavelet transformation was used for the bearing-fault identification and classification.

ACTIVE VIBRATION CONTROL OF COMPOSITE PLATE USING AFC ACTUATOR AND PVDF SENSOR

A finite element formulation for active vibration control of laminated composite plate integrated with active fiber composite (AFC) layer acting as distributed actuator and PVDF layer as sensor is presented in this paper. An eight noded quadratic isoparametric element with five mechanical degrees of freedom and one electrical degree of freedom per node of the element is considered. Newmark time integration method is used to calculate the dynamic response and negative velocity feedback control algorithm is used to control the dynamic response of the laminated composite plate. The effect of piezoelectric fiber orientation in actuator layer on the control of vibration is studied.

Fundamental accuracy of time domain finite element method for sound-field analysis of rooms

This paper presents an assessment of the accuracy and applicability of a time domain finite element method (TDFEM) for sound-field analysis in architectural space. This TDFEM incorporates several techniques: (1) a hexahedral 27-node isoparametric acoustic element using a spline function; (2) a lumped acoustic dissipation matrix; and (3) Newmark time integration method with an absolute diagonal scaled COCG iterative solver. Sound fields in an irregularly shaped reverberation room of 166 m 3 are computed using TDFEM. The computed values and measured values for 125–500 Hz are compared, revealing that the fine structure of the computed band-limited impulse responses agree with measured ones up to 0.1 s, with a cross-correlation coefficient greater than 0.93. The cross-correlation coefficient decreases gradually over time, and more rapidly for higher frequencies. Moreover, the computed decay curves, and the reverberation times, agree well with the respective measured ones, and with a better fit the higher the frequency (up to 500 Hz).

Dynamic analysis for a pair of spur gears with translational motion due to bearing deformation

In this study, the dynamic response of a pair of spur gears is analyzed when the gear set has translational motion due to bearing deformation. A new dynamic model for the gear set, considering translational motion, is proposed, in which the distance between the centers of a pinion and a gear varies with time. Therefore, the proposed model regards the pressure angle and the contact ratio as time-varying variables, while the previous model regards them as constants. After deriving nonlinear equations of motion for the gear set, the dynamic responses are computed by applying the Newmark time integration method. This paper claims that the new model produces more accurate dynamic responses in comparison to those of the previous model. Some dynamic response differences between the new and previous models are demonstrated, and the effects of damping and stiffness upon the dynamic responses are also investigated.

공극의 시간변화를 고려한 3 상 BLDC 모터의 동특성 해석

In this study, vibrations of an electric motor are analyzed when the motor has the interaction between mechanical and electromagnetic behaviors. For this vibration analysis a 3-phase 8-pole brushless DC motor is selected. Vibrations of the motor are influenced by coupled electromechanical characteristics. The variation of air-gap induced by vibration has an influence on the inductance of the motor coil. To analyze dynamic characteristics of the rotor, we studied inductance by the variation of an air-gap. After obtaining the kinetic, potential and magnetic energies for the motor, the equations of motion are derived by using Lagrange's equation. By applying the Newmark time integration method to the equations, the dynamic responses for the displacements and currents are computed.

Simulation of fractionally damped mechanical systems by means of a Newmark-diffusive scheme

A Newmark-diffusive scheme is presented for the time-domain solution of dynamic systems containing fractional derivatives. This scheme combines a classical Newmark time-integration method used to solve second-order mechanical systems (obtained for example after finite element discretization), with a diffusive representation based on the transformation of the fractional operator into a diagonal system of linear differential equations, which can be seen as internal memory variables. The focus is given on the algorithm implementation into a finite element framework, the strategies for choosing diffusive parameters, and applications to beam structures with a fractional Zener model.

Nonlinear numerical simulation method for galloping of iced conductor

Based on the principle of virtual work, an updated Lagrangian finite element formulation for the geometrical large deformation analysis of galloping of the iced conductor in an overhead transmission line is developed. In numerical simulation, a threenode isoparametric cable element with three translational and one torsional degrees-offreedom at each node is used to discretize the transmission line. The nonlinear dynamic system equation is solved with the Newmark time integration method and the Newton-Raphson nonlinear iteration. Numerical examples demonstrate the efficiency of the presented method and the developed finite element program. A new possible galloping mode, which may reflect the saturation phenomenon of a nonlinear dynamic system, is discovered under the condition that the lowest order of vertical natural frequency of the transmission line is approximately two times of the horizontal one.

A non‐linear coupled finite element–boundary element model for the prediction of vibrations due to vibratory and impact pile driving

AbstractThis paper presents a non‐linear coupled finite element–boundary element approach for the prediction of free field vibrations due to vibratory and impact pile driving. Both the non‐linear constitutive behavior of the soil in the vicinity of the pile and the dynamic interaction between the pile and the soil are accounted for.A subdomain approach is used, defining a generalized structure consisting of the pile and a bounded region of soil around the pile, and an unbounded exterior linear soil domain. The soil around the pile may exhibit non‐linear constitutive behavior and is modelled with a time‐domain finite element method. The dynamic stiffness matrix of the exterior unbounded soil domain is calculated using a boundary element formulation in the frequency domain based on a limited number of modes defined on the interface between the generalized structure and the unbounded soil. The soil–structure interaction forces are evaluated as a convolution of the displacement history and the soil flexibility matrices, which are obtained by an inverse Fourier transformation from the frequency to the time domain. This results in a hybrid frequency–time domain formulation of the non‐linear dynamic soil–structure interaction problem, which is solved in the time domain using Newmark's time integration method; the interaction force time history is evaluated using the θ‐scheme in order to obtain stable solutions.The proposed hybrid formulation is validated for linear problems of vibratory and impact pile driving, showing very good agreement with the results obtained with a frequency‐domain solution. Linear predictions, however, overestimate the free field peak particle velocities as observed in reported field experiments during vibratory and impact pile driving at comparable levels of the transferred energy. This is mainly due to energy dissipation related to plastic deformations in the soil around the pile. Ground vibrations due to vibratory and impact pile driving are, therefore, also computed with a non‐linear model where the soil is modelled as an isotropic elastic, perfectly plastic solid, which yields according to the Drucker–Prager failure criterion. This results in lower predicted free field vibrations with respect to linear predictions, which are also in much better agreement with experimental results recorded during vibratory and impact pile driving. Copyright © 2008 John Wiley & Sons, Ltd.

유동점성 및 공탄성 변형효과를 고려한 스테이터-로터 케스케이드 시스템의 성능평가

본 연구에서는 블레이드 구조 변형 효과를 고려하여 스테이터-로터 상호간섭 케스케이드 모델의 성능평가를 위한 유체-구조 연계해석 시스템을 개발하였다. 고정된 스테이터와 회전하는 로터는 상호간섭 영향이 유동해석에 고려되었으며, 레이놀즈-평균화 난류 방정식인 Spalart-Allmaras 모델과 k-ω SST 난류 모델이 압축성 유동박리 효과를 고려한 유동하중을 예측하기 위해 적용되었다. 정적인 유체-구조 연계해석과 수렴율 증진을 효과적으로 수행하기 위하여 큰 인공 감쇠를 가지는 연계 Newmark 시간적분 기법을 적용하였다. 수치실험을 통해 탄성축 위치에 따른 구조변형 효과가 케스케이드 성능에 미치는 영향을 파악하였다. 구조변형 효과가 고려된 경우 일반적인 강체 블레이드 모델에 대한 성능예측 결과와 다소 차이가 유발될 수 있음을 보였으며 공력탄성학적 영향을 고찰하였다. In this study, advanced (fluid-structure interaction (FSI)) analysis system has been developed in order to predict turbine cascade performance with blade deformation effect due to aerodynamic loads. Intereference effects due to the relative movement of the rotor cascade with respect to the stator cascade are also considered. Reynolds-averaged Navier-Stokes equations with one equation Spalart-Allmaras and two-equation k-ω SST turbulence models are solved to accurately predict fluid dynamic loads considering flow separation effects. A fully implicit time marching scheme based on the (coupled Newmark time-integration method) with high artificial damping is efficiently used to compute the complex fluid-structure interaction problem. Predicted aerodynamic performance considering structural deformation effect of the blade shows somewhat different results compared to the case of rigid blade model. Cascade performance evaluations for different elastic axis positions are importantly presented and its aeroelastic effects are investigated.

Dynamic Behavior of Reinforced Concrete Beams Strengthened with Composite Materials

The dynamic behavior of reinforced concrete (RC) beams strengthened with externally bonded composite materials is analytically investigated. The analytical model is based on dynamic equilibrium, compatibility of deformations between the structural components (RC beam, adhesive, composite material) and the concept of the high order approach. The equations of motion along with the boundary and continuity conditions are derived using Hamilton’s variational principle and the kinematic relations of small deformations. The mathematical formulation also includes the constitutive laws that are based on beam and lamination theories, and the two-dimensional elasticity representation of the adhesive layer including the closed form solution of its stress and displacement fields. The Newmark time integration method, which is directly applied to the resulting set of coupled partial differential equations, is adopted. This procedure yields a set of ordinary differential equations, which are analytically or numerically solved in every time step. The response of a strengthened beam to different dynamic loads that include impulse load, harmonic load, and seismic base excitation is numerically investigated. The numerical study highlights some of the phenomena associated with the dynamic response and explores the capabilities of the proposed model. The paper closes with a summary and conclusions.

Active damping of rotating composite thin-walled beams using MFC actuators and PVDF sensors

The object of this research is to enhance the damping performance for vibration suppression of rotating composite thin-walled beams using MFC actuators and PVDF sensors. The formulation is based on single cell composite beam including a warping function, centrifugal force, Coriolis acceleration and piezoelectric effect. Adaptive capability of the beam is acquired through the use of a negative velocity feedback control algorithm. Numerical analysis is performed using finite element method and Newmark time integration method is used to calculate the time response of the model. It is observed that the feedback control gain has an effect on damping performance. The paper continues with an investigation into influences of parameters such as the rotating speed and the fiber orientation in host structures. Also, it is confirmed that effective damping performance is achievable through the suitable arrangement and distributed size of sensor and actuator pair using case study.

Resolution of Defects in Degenerated Shell Elements Through Modification of Gauss Integration

In this paper, the modified Gauss integration method is developed to eliminate the shear and membrane locking phenomena of the degenerated shell element. The behavior of the element based on the Mindlin/Reissner theory in plates and shells sometimes causes a problem. In displacement-based shell elements, the full integration of stiffness matrices leads to a 'locking' or over-stiff behavior. The selective or reduced integration procedures may often overcome these difficulties, while overstiff solutions may still occur in the analysis with a highly constrained boundary. Except for the six zero-energy modes associated with shell rigid body movements, during the reduced integration of the stiffness matrices, many extra zero spurious energy modes are introduced due to reduced integration. This is a serious defect of degenerated shell element. In previous studies, several methods such as the addition of nonconforming displacement modes, an assumed strain method, and hybrid and mixed elements have been introduced in an attempt to overcome these difficulties. In this study a newly modified Gauss integration method combining both a selective and a weight-modified integration is suggested. Numerical experiments show that the new selective integration and weight-modified integration rule is very effective in eliminating the shear and membrane locking in static and modal analyses, and removes spurious zero-energy modes as well. Also, the effectiveness of proposed shell element is tested by applying it to some example problems. We solved post-buckling problem by the Riks arc-length method and dynamic problem by the Newmark's time integration method, as well as static problems.