Precise Integration Method Research Articles

Improved precise integration method for chatter stability prediction of two-DOF milling system

The motivation of this paper is to update the precise integration method (PIM) by a second-order Taylor formula and make detailed contrasts with the existing PIM, the semi-discretization method (SDM) to exhibit the necessity of developing this improved PIM (IPIM). The dynamics of two-degree of freedom (DOF) milling process with consideration of regeneration effect is first governed by a time periodic delay differential equation (DDE). With time period being evenly divided into a limited set of intervals, the integral non-homogeneous element is approximated by the second-order Taylor formula in every small time segment. After decomposing the exponential factor into a real term with 2N order algorithm, the transition matrix representing the specific machining system state is established in one whole tooth passing period to search for the chatter-free borderline. To investigate the characteristics of the proposed method in convergence rate, prediction accuracy, and computational efficiency, the benchmark example used in the literatures is introduced to develop a battery of comparisons with PIM and SDM. Finally, the experimental verification is also conducted in a CNC machine tool to further confirm the operability of the proposed IPIM, and the results indicate the method is of availability.

Reliable nonlinear hybrid simulation using modified operator splitting technique

One of the main challenges of hybrid simulation is developing integration methods that not only provide accurate and stable results but also are compatible with the hybrid simulation circumstances. This paper presents a novel enhanced integration technique for hybrid simulation termed “modified operator splitting” (MOS) method. The main aim of the MOS technique is to improve the precision of the operator splitting (OS) method by reducing the corrector step length, where initial stiffness is utilized instead of actual stiffness. For this purpose, a new algorithm is proposed, which makes a more precise estimation of the predictor displacement; thus minimizes the effect of the corrective procedure. As a result, a higher percentage of the restoring force is based on actual experimental behavior rather than being approximated, which finally leads to a very precise and reliable integration method, especially for nonlinear hybrid simulations. Performance of the MOS method is evaluated by the following: (a) Analytical studies, (b) numerical simulations, and (c) hybrid simulation. The accuracy analysis for a wide range of structures and ductility levels demonstrates the superior precision of MOS over OS, especially for severe nonlinearities and larger Δt/T0. In terms of stability, it is shown that for practical applications of civil engineering problems, MOS provides stable results. Furthermore, the application of MOS to hybrid simulation not only again verifies its higher precision over OS, but also shows that MOS minimizes the accumulation of errors during the test; the characteristic that is desirable for a method applied to feedback systems such as hybrid simulation.

Thermo-mechanical performance of layered transversely isotropic media around a cylindrical/tubular heat source

We develop a new numerical model based on a precise integration method to investigate the coupled thermo-mechanical performance of layered transversely isotropic media around a cylindrical/tubular heat source. To obtain the relational matrices of the extended precise integration method, we first convert the governing equations of the problem into ordinary differential matrix equations through the Laplace–Hankel transform. Then, the cylindrical heat source is divided into a series of plane heat sources, and the plane temperature load term is added to the state vector between layer elements. By combining the layer elements, we build a layered transversely isotropic numerical model containing a cylindrical heat source in the transformed domain. Finally, we solve the model in the transformed domain and obtain the solution of the problem in the real domain through the Laplace–Hankel transform inversion. The accuracy of this method is verified by comparing the solutions with the results of the analytical method and the finite element method. Then, we study the influence of the anisotropy of thermal parameters, the embedded depth, the length/radius ratio, the type of heat source and the stratification of the medium on the thermo-mechanical coupled performance.

An efficient method for simulating the dynamic behavior of periodic structures with piecewise linearity

An efficient method based on the parametric variational principle (PVP) is proposed for simulating the dynamic behavior of periodic structures with large number of piecewise linearity. The formulation for the gap-activated springs is proposed based on PVP, which can accurately determine the states of the gap-activated springs. Based on the periodicity of the system and the precise integration method, an efficient numerical time-integration method is developed to obtain the dynamic responses of the system. For this method, the matrix exponential of only one unit cell of the system is computed, which greatly improves the computational efficiency. Dynamic responses of a 3 degrees of freedom (DOF) piecewise linear system under harmonic excitations are given to demonstrate the validation of the proposed method. The piecewise linear dynamic system can exhibit very complex vibrational behavior, such as stable periodic motion, multi-periodic motion, quasi-periodic motion and chaotic motion, which can be successfully predicted by using bifurcation theory. Moreover, it is demonstrated that the proposed method can be used to efficiently determine dynamic responses of a periodic piecewise linear system with large number of DOFs and large number of gap-activated springs under harmonic excitations.

Highly precise time integration method for linear structural dynamic analysis

SummaryTo achieve high accuracy, conventional time integration methods require small time steps, especially when applied to large‐scale finite element systems. The use of small time steps calls for a large number of computations. In this paper, a strategy is proposed to construct an efficient and highly precise time integration method for linear time‐invariant systems. Adopting the multi‐substep notion, the highly precise time integration method creates an integrated amplification matrix prior to recursive calculations by multiplying the amplification matrices of N equal substeps, where a 2m algorithm and the method of storing incremental matrix are respectively employed to reduce computational cost and rounding error. On the basis of the Newmark method, two highly precise symplectic schemes and one highly precise dissipative scheme are developed. In addition, the consistent stability of the used Newmark schemes is presented. The free vibration of a three‐dimensional truss, the impact of a particle against a rod, and the forced vibrations of a rectangular thin plate and a stiff system are investigated to validate the performance of the proposed method.

The precise integration method for semi-discretized equation in the dual reciprocity method to solve three-dimensional transient heat conduction problems

A dual reciprocity precise integration method (DRPIM) is presented to solve 3-dimensional transient heat conduction problems. In the presented method, the transient heat conduction problem, which is an initial boundary value problem, is transformed into an initial value problem (IVP) through a dual reciprocity scheme. Then an analytical solution, which is described by terms of the matrix exponential function (MEF), to this IVP is applied. A precise integration method (PIM) is finally applied to compute the MEF accurately. The accuracy and stability of the presented method are demonstrated by three numerical examples.

Precise Time Integration Methods for Transient Response Analysis of Large-Scale Structures

To remove the calculations of inverse and/or exponential of large-scale matrix in the precise time integration (PTI) method for transient response analysis of structures, new PTI methods equipped with the restarted Krylov subspace method and real Leja points interpolation method to calculate are presented in this paper. In both kinds of methods, only the multiplications of sparse matrix and vectors are involved. The theories and implementations of the improved PTI methods are introduced. The transient responses of a cantilever beam and an aeroengine bladed disk are used as numerical examples to validate the accuracy and efficiency of the proposed methods. The effects of tunable parameters of both kinds of algorithms on the numerical efficiency are also discussed. It has been demonstrated that the improved PTI methods are accurate and insensitive to integration step length and, more importantly, are highly efficient for large-scale structures subjected to high-frequency excitations, which greatly extend the applicability of PTI methods.

A Modified Precise Integration Method for Long-Time Duration Dynamic Analysis

This paper presents a modified Precise Integration Method (PIM) for long-time duration dynamic analysis. The fundamental solution and loading operator matrices in the first time substep are numerically computed employing a known unconditionally stable method (referred to as original method in this paper). By using efficient recursive algorithms to evaluate these matrices in the first time-step, the same numerical results as those using the original method with small time-step are obtained. The proposed method avoids the need of matrix inversion and numerical quadrature formulae, while the particular solution obtained has the same accuracy as that of the homogeneous solution. Through setting a proper value of the time substep, satisfactory accuracy and numerical dissipation can be achieved.

Vibration of initially stressed carbon nanotubes under magneto-thermal environment for nanoparticle delivery via higher-order nonlocal strain gradient theory

In this paper, the forced vibration of a single-walled carbon nanotube (SWCNT) under a moving nanoparticle is investigated based on the higher-order nonlocal strain gradient theory. The SWCNT is subjected to thermo-mechanical stresses and an external longitudinal magnetic field. The influences of higher-order stress gradients in conjunction with the strain gradient nonlocality are taken into account. Using Hamilton’s principle and Maxwell’s equations, the higher-order differential equations of motion are derived. An analytical solution is obtained for the dynamic deflection of SWCNTs using the Galerkin method. Furthermore, the governing differential equation is solved numerically using the precise integration method. The results of the two solution procedures are compared and an excellent agreement is found between them. Finally, the influences of various scale parameters, the velocity of the moving nanoparticle, the initial axial stress, the temperature change and longitudinal magnetic field on the dynamic response of SWCNTs are investigated.

Precise integration method for natural frequencies and mode shapes of ocean risers with elastic boundary conditions

• A relatively new precise integration method for modal analysis of ocean risers is developed. • It is suitable for ocean risers with variable tension and cross-section and various boundaries. • The results are proved to be very accurate by comparing the analytical solution and literature. • Computing time is quite short and the method exhibits good convergence and high efficiency. • Effects of elastic constraints simulating damaged and undamaged boundaries on natural frequencies and mode shapes are investigated. The effects of the elastic constraints simulating damaged and undamaged boundaries on the natural frequencies and mode shapes of ocean risers with a variable axial tension are investigated using the precise integration method (PIM). The classical high-order variable-coefficient partial-differential governing equation of the free vibration of ocean risers is reduced to a set of first-order ordinary differential equations and efficiently solved by the PIM. The main advantages of the PIM are that the numerical results can be calculated with high accuracy even when total element number n , number of iterations N and Taylor expansion terms r are small. Moreover, the computing time is quite short. Various boundary conditions are modeled as linear elastic constraints using a pair of translational and torsional springs, and four types of boundary damage coefficients are proposed to investigate the effects of a damaged boundary on the natural frequencies. The results for specific boundary conditions show agreement with those reported in the literature, and the calculation errors are very small in comparison with the analytical solution. Overall, the methodology of PIM is applicable for the investigation of the natural frequencies and mode shapes of ocean risers with a variable axial tension and cross-section and various boundary conditions.

An accurate method for the dynamic behavior of tensegrity structures

PurposeThe purpose of this paper is to propose an accurate and efficient numerical method for determining the dynamic responses of a tensegrity structure consisting of bars, which can work under both compression and tension, and cables, which cannot work under compression.Design/methodology/approachAn accurate time-domain solution is obtained by using the precise integration method when there is no cable slackening or tightening, and the Newton–Raphson scheme is used to determine the time at which the cables tighten or slacken.FindingsResponses of a tensegrity structure under harmonic excitations are given to demonstrate the efficiency and accuracy of the proposed method. The validation shows that the proposed method has higher accuracy and computational efficiency than the Runge–Kutta method. Because the cables of the tensegrity structure might be tense or slack, its dynamic behaviors will exhibit stable periodicity, multi-periodicity, quasi-periodicity and chaos under different amplitudes and frequencies of excitation.Originality/valueThe steady state response of a tensegrity structure can be obtained efficiently and accurately by the proposed method. Based on bifurcation theory, the Poincaré section and phase space trajectory, multi-periodic vibration, quasi-periodic vibration and chaotic vibration of the tensegrity structures are predicted accurately.

Time-dependent behavior of axisymmetric thermal consolidation for multilayered transversely isotropic poroelastic material

The governing equations for axisymmetric thermal consolidation of transversely isotropic poroelastic material are treated by employing Laplace–Hankel transforms to deduce the standard differential equations. Then the extended precise integration method is applied to solve the above standard differential equations. Combining with the continuity conditions and the boundary conditions, the solution in the transformed domain can be achieved. The actual solution can be acquired by inverting the integral transformations. Several numerical examples are provided to examine the accuracy of the theory and to discuss the effects of the properties of materials and the stratification on the time-dependent behavior of thermal consolidation for multilayered transversely isotropic poroelastic material.

Coupled Dynamic Characteristics of Wind Turbine Gearbox Driven by Ring Gear Considering Gravity

Gravity is usually neglected in the dynamic modeling and analysis of the transmission system, especially in some relatively lightweight equipment. The weight of wind turbine gearbox is up to tens of tons or even hundreds of tons, and the effects of gravity have not been explored and quantified. In order to obtain accurate vibration response predictions to understand the coupled dynamic characteristics of the wind turbine gear transmission system, a comprehensive, fully coupled, dynamic model is established by the node finite element method with gravity considered. Both time-domain and frequency-domain dynamic responses are calculated using the precise integration method with various excitations being taken into account. The results indicate that gravity has a significant impact on the vibration equilibrium position of central floating components, but the changing trends are different. Gravity does not change the composition of the excitation frequency, but will have a certain impact on the distribution ratio of the frequency components. And the high frequency vibrations are hardly affected by gravity. In addition, the load sharing coefficient is greater when gravity is taken into account, both of internal gearing and of external gearing system. When the planet gears have a certain position error in accordance with certain rules, the load sharing performance of the system will be better.

A comparative study of design sensitivity analysis based on adjoint variable method for transient response of non-viscously damped systems

Abstract In this paper, design sensitivity analysis methods for the transient response of non-viscously damped systems are considered. The damping forces of the non-viscously damped systems depend on the past history of motion via convolution integrals over some suitable kernel functions. The adjoint variable method (AVM) is adopted to develop the design sensitivity analysis. Two numerical solution schemes, namely the discretize-then-differentiate method and the differentiate-then-discretize method, are introduced to complete the AVM for the sensitivity analysis of non-viscously damped systems. The discretize-then-differentiate AVM discretizes the equations of motion based on the Newmark- β implicit integration method first and then differentiates the discrete equations. On the contrary, the differentiate-then-discretize AVM firstly differentiates the equations of motion after transforming it into a state-space form, and then discretizes the equations based on a modified precise integration method (PIM). The numerical accuracy, efficiency, consistency and implementation effort are discussed and compared. Two numerical examples are presented to show the effectiveness of both methods. The results indicate that, by considering both computational accuracy and efficiency, the PIM based differentiate-then-discretize method is more suitable than the Newmark- β based discretize-then-differentiate method for the sensitivity analysis of transient responses for non-viscously damped systems.

Estimation of boundary condition on the furnace inner wall based on precise integration BEM without iteration

This study develops a non-iterative method for estimating the boundary condition of transient heat conduction problems on the furnace inner wall based on the precise integration boundary element method (PIBEM). Generally, the problem of inversing boundary condition needs the iteration direct process about many times. The computational cost is high especially for inversing the transient boundary condition. However, the present method can directly inverse the temperature and heat flux boundary conditions by the least-square error method without iteration. In addition, using the precise integration method and BEM can fully play the superiority of solving the time-domain problem and reducing the dimension of problem, respectively. Meanwhile, an expanding coefficient method is introduced to approximate the unknown boundary condition. Numerical results show that the steady and accurate results can be obtained by the approximation of quadratic polynomials. A number of examples validate that the present method has very high efficiency and precision by investigating the effect of many factors such as the measurement error, the position and number of measurement points.

The semi-analytical method for damping of tubular transition layer damping structure

To solve the limited vibration consumption of the traditional tubular damping structure (TTDS), the tubular transition layer damping structure (TTLDS) is proposed; Based on viscoelastic materials and theories of thin cylindrical shells, the governing equation, the first order matrix differential equation describing vibration of TTLDS under harmonic excitation, is derived by considering the interaction between all layers and the dissipation caused by the shear deformation for transition layer and damping layer. By using the extended homogeneous capacity precision integration method to solve the control equation, a semi-analytical method for studying the vibration and damping characteristics of TTLDS is given. By way of comparison, the correctness of the method provided in paper is verified. At last, the influence of thickness, material and location of transition layer on damping effect is analyzed. The results show that the change for the thickness or material of the transition layer can make the structural damping effect change greatly, while the change for location of the transition layer plays only a few roles on the structural damping effect.

An interval precise integration method for transient unbalance response analysis of rotor system with uncertainty

Abstract A non-intrusive interval precise integration method (IPIM) is proposed in this paper to analyze the transient unbalance response of uncertain rotor systems. The transfer matrix method (TMM) is used to derive the deterministic equations of motion of a hollow-shaft overhung rotor. The uncertain transient dynamic problem is solved by combing the Chebyshev approximation theory with the modified precise integration method (PIM). Transient response bounds are calculated by interval arithmetic of the expansion coefficients. Theoretical error analysis of the proposed method is provided briefly, and its accuracy is further validated by comparing with the scanning method in simulations. Numerical results show that the IPIM can keep good accuracy in vibration prediction of the start-up transient process. Furthermore, the proposed method can also provide theoretical guidance to other transient dynamic mechanical systems with uncertainties.

Quasi-static thermal analyses of layered compressible poroelastic materials with a finite depth or half-space

• The Laplace–Fourier transforms are applied to reduce partial differential equations to ordinary ones. • An extended precise integration solution of multilayered materials due to a heat source is derived. • The actual solution in the physical domain is further solved by inverting the Laplace-Fourier transforms. • The effects of compressible constituents and layered characteristic on thermal consolidation are investigated. Quasi-static thermal analyses of layered compressible poroelastic materials with a finite depth or half-space caused by a heat source are carried out via extending the precise integration method in this study. The Laplace-Fourier transforms are applied to reduce partial differential equations of three-dimensional coupled thermo-hydro-mechanical problem to ordinary ones in the transform domain. By means of the precise integration method, a precise solution of layered materials induced by a heat source is deduced, and then, the previous ordinary differential equations are solved by substituting these equations into the extended precise integration solution. The actual solution is solved by applying the Laplace–Fourier inverse transforms. And lastly, numerical computations are performed to validate the method, and to analyze the effects of compressible constituents and layered characteristic as well as the heat source's half-life on thermal consolidation.

Chatter stability prediction for five-axis ball end milling with precise integration method

The differential dynamic equation (DDE) considering regenerative chatter effect is derived from an ideal cutting model. At the same time, the ball end mill-workpiece engagement (BWE) in five-axis milling is efficiently extracted by the semi-analytical method, where the machining residual left on the finished surface is taken into account. The explicit precise integration method (PIM) is adopted to translate the DDE into a time series expression, ensuring stability prediction without any discretization error loss. With the relationship of cutter edge and BWE at different instants, the stability lobe diagram (SLD) is constructed by Floquet theory, which is subsequently testified in the five-axis NC machine tool. The experimental results keep good agreement with estimations, validating the proposed method. Compared with traditional method in dynamic cutting forces modeling, the proposed approach is more close to actual processing. In contrast to Runge-Kutta based complete discretization method, PIM shows higher superiority no matter in convergence rate or calculation accuracy. Lastly, the influences of different inclination angles on stability are revealed, which can give a reference to selection of suitable process parameters for higher productivity.

Efficient and accurate method for 2D periodic structures based on the physical features of the transient heat conduction

An efficient and accurate method is developed to solve the transient heat conduction problem in two-dimensional (2D) periodic structures. For a 2D periodic structure, according to the physical features of the transient heat conduction, the periodic property of the structure, and the physical meaning of the matrix exponential, it is demonstrated that the matrix exponential for a reasonable time step is a sparse matrix containing many identical elements. Next, based on the superposition principle of linear systems and the algebraic structure of the matrix exponential, computation of the response of original 2D periodic structure is transformed into computation of the responses of small-scale models with several unit cells. Finally, the precise integration method (PIM) is used to compute the temperature responses of the small-scale models. The proposed method not only inherits the accuracy and stability of the PIM but also achieves significantly improved computational time and storage requirements. A series of numerical examples demonstrate that the proposed method is more efficient than the Crank-Nicholson method and provides highly precise solutions even with a larger time step.