Constant Average Acceleration Method Research Articles

Exciters of vibrations with two pairs of impacting surfaces

Exciters of vibrations of the type when one vibrating mass performs impacts into two immovable supports, which are located on the opposite sides with respect to the mass, are investigated. Here because of the connection of the vibrating mass with the supports the system may have the point of equilibrium located at equal or different distances from the impacting surfaces. Also, because of the difference of coefficients of restitution of impacts the system may have symmetric or non-symmetric laws of motions of the vibrating mass. Such systems are important in pipe robots and other engineering devices. Typical results of investigation of dynamics of such systems are obtained. The presented results can be used in the process of design of systems with vibrators having two impacting pairs. Investigations are performed by using analytical as well as numerical methods (calculations are performed by C++ Builder Community Edition, Newmark constant average acceleration method is used for numerical integration of equations of motion). The obtained typical graphical relationships show symmetric as well as unsymmetric regimes of motion and enable to perform their comparison. The basic novelty of this paper is the investigation of unsymmetric regimes of motion.

A time-domain integration method with equivalent complex damping model based on viscoelastic material constitutive relation

The complex damping model is only applied in frequency-domain. Based on the constitutive relation of viscoelastic materials, a vibration equation is equivalent to the constitutive relation of complex damping theory and the frequency response function satisfying the causality constraint. Compared with the existing equivalent complex damping model, the proposed equivalent damping model can consider more natural frequencies. The calculation formulas of Gauss precise integral method and improvement constant average acceleration method are derived. By the equivalent complex damping, the seismic time-history response of the typical example could carry on the numerical calculation. Compared with the Gauss precise integral calculation results of complex damping vibration equation, some conclusions can be analyzed. The proposed two methods could avoid divergent phenomenon in the time-domain numerical solution of complex damping vibration equation. The time-domain integral method of equivalent complex damping theory is stable and convergent. The Gauss precise integral method has strong equivalence and big computational complexity. The improvement constant average acceleration method has weak equivalence and small computational complexity.

Numerical analysis of non-Fourier heat conduction dynamics in the composite layer

This paper presents the numerical analysis of non-Fourier heat conduction in thin composite layers under asymmetrical boundary conditions. In the thermal barriers such as steam and gas turbine blades, thin film coating are used to protect the blade from thermal damage. The coating on the blades are very short in length. Heat conduction across thin composite layer with short time is examined using a finite element approach. With this very small duration with the finite speed of the thermal wave, the Fourier mode of heat conduction is disappeared due to the infinite speed of the thermal wave assumption. Therefore, analyzing the non-Fourier heat conduction in thin layers is essential. The developed model is executed in Python using Newmark's scheme and the constant average acceleration method to predict the temperature variation and temperature contours. The present model is validated with an experimental and numerical solution with good agreement. Besides, the temperature distribution across the composite layer with the entire length of the substrate and the coating for different thermal conductivity values, thermal diffusivity, and relaxation time are examined. It is noted that when the dimensionless

Numerical investigation of transient thermo-mechanical loading effect on combined damaged (crack and delamination) curved shell structure: An experimental verification

The transient responses of the curved shell panels are examined under thermo-mechanical loading and verified experimentally. The damaged shell model is formulated mathematically considering polynomials of higher-order based on isoparametric finite element (FE) steps. The dynamic (time-dependent deflection) responses are evaluated numerically using developed computer code in MATLAB, utilizing Newmark's constant average acceleration method. The accuracy of the present model is proven by equating the consequences with the available data in the published literature. Additionally, a few experimentations have been accomplished and outcomes are equated with the numerical solution to show the effectiveness of the present developed numerical model. Further, the impact of damages (debond and/or crack), numerous structural input parameters (curvature ratio, shell geometry) and temperature on the curved shell panel is demonstrated and presented through a wide variety of numerical examples.

A fully explicit staggered algorithm for near‐field wave propagation of fluid‐saturated porous media based on u‐p dynamic formulation

AbstractNear‐field wave propagation of the fluid‐saturated porous media (FSPM) is an important issue in soil dynamics and geotechnical earthquake engineering. In the current study, a fully explicit staggered algorithm for the near‐field wave propagation analysis of the FSPM in the time domain is developed based on the u‐p dynamic formulation. The decoupling of dynamic formulation is implemented by the diagonalization of the mass matrix and pore fluid compressibility matrix. The central difference method and Newmark constant average acceleration method are used for the solution of the displacement and velocity of solid skeleton, and the backward difference method is used for the solution of pore fluid pressure in the time domain. The effectiveness and feasibility of the matrix diagonalization are verified by the comparison with the classical numerical algorithm. The proposed algorithm is also validated by the comparison of the numerical results with the corresponding analytical solution results. Combing with the transmitting artificial boundary condition, the proposed algorithm is then applied to investigate the seismic response of a FSPM free field, and its applicability to the typical near‐field wave propagation problems of the FSPM is indicated. The stability analysis of the proposed algorithm is finally conducted, and the corresponding stability criterion is presented. For the proposed algorithm, all the dynamic variables of the FSPM are solved in an iterative way and the coupling dynamic equations need not to be solved at each time step. The computational effort can be reduced considerably, and the computational efficiency can be improved remarkably.

High-order implicit time integration scheme based on Padé expansions

A single-step high-order implicit time integration scheme for the solution of transient as well as wave propagation problems is presented. It is constructed from the Padé expansion of the matrix exponential solution of a system of first-order ordinary differential equations formulated in the state–space. By exploiting the techniques of polynomial factorization and partial fractions of rational functions, and by decoupling the solution for the displacement and velocity vectors a computationally efficient scheme is developed. An important feature of the novel algorithm is that no direct inversion of the mass matrix is required. Based on the diagonal Padé expansion of order M a time-stepping scheme of order 2M is constructed. When M is odd, one system of equation with a real matrix and (M−1)/2 systems of equations with complex matrices are solved recursively. When M is even, there are M/2 systems of equations with complex matrices. These systems are sparse and comparable in complexity to the standard Newmark method, i.e., the effective system matrix is a linear combination of the static stiffness, damping, and mass matrices. The derived second-order scheme is analytically equivalent to Newmark constant average acceleration method. The proposed time integrator has been implemented in MATLAB and FORTRAN using direct linear equation solvers. In this article, numerical examples featuring nearly one million degrees of freedom are presented to highlight the exceptional accuracy and efficiency of the novel time-stepping scheme in comparison with established second-order methods. The MATLAB- and FORTRAN-implementations are available from the authors upon request.

Three-dimensional dynamic behaviour of flexible catenary risers with an internal slug flow

This work analyses the three-dimensional dynamic response of flexible catenary risers when subjected to the effects of an internal slug flow. The interaction between the flexible riser and the internal slug flow is a rather complex phenomenon, and this is because the forces due to the acceleration of the internal fluid influence the riser dynamics, which, in turn, influences the slug flow motion in its interior. In order to carry out this analysis, a new computational tool called SLUGFLEX has been developed, which is composed by two computational codes: the code that calculates the structural dynamic response of the flexible riser, and the other one that predicts the development of the internal slug flow. In the structural analysis, the spatial discretization of the riser is performed using spatial Euler beam finite elements, which suffer large displacements, although with small deformations. The tangent stiffness matrix and the internal force vector of the structure are calculated using the corotational approach. The internal slug flow, whose key feature is the intermittence of two different regions: a liquid slug and a large bubble of gas, is mathematically modelled by using a new one-dimensional lagrangian slug tracking method, which has been based on a model developed for straight pipes, and over which the effects of the curvature of the riser and its dynamics were added. The dynamic response of the riser is calculated by using the constant average acceleration method, in combination with the incremental iterative Newton–Raphson scheme for follower loads. The numerical results show good agreement when compared to the experimental ones obtained from the literature.

Numerical simulation of thermal wave propagation and collision in thin film using finite element solution

Numerical simulation of propagation and collision of thermal wave phenomena in the order of nanometer is carried out. The finite element procedure is developed for Cattaneo–Vernotte heat conduction equation. In this simulation, non-homogeneous boundary conditions are used. Newmark’s scheme and constant average acceleration method are used to solve second-order time derivatives. The non-dimensional results are compared with the analytical solutions of the same kind of problem solved by using superposition principle with solution structure theorems and experimental values. It is noted that this current procedure had good agreement with the results given by analytical and experimental solutions. In this study, the finite element modeling of hyperbolic equation is solved for thermal wave propagation and collision of thermal waves. The transition of hyperbolic to parabolic nature of thermal wave propagation for different layer lengths and the characteristics of nodal temperatures near to the boundary are discussed. Also this work examined the temperature variation at the center of the layer at all collisions, which shows the increase in temperature after the first collision when time is 0.5 and then propagates toward boundary with further increase in temperature. This is well above the applied boundary temperature and decrease in temperature at boundary for subsequent collisions. The transition of thermal waves from hyperbolic to parabolic nature is analyzed and is found that the transition happens for the film thickness greater than 5. This phenomenon is repeated until it reaches steady state when time is 5 for hyperbolic whereas 1.5 for parabolic mode.

Nonlinear Dynamic Response and Stability Analysis of a Tensegrity Bridge to Selected Cable Rupture

This paper presents a nonlinear dynamic analysis procedure used for the investigation of the response of a tensegrity bridge to a selected sudden cable rupture. In order to simulate a cable rupture, for the loaded or unloaded geometry of the tensegrity structure, a geometrical nonlinear analysis is performed, and the cable end tensions projected in the global coordinate system are determined. Next, these forces are applied as external nodal forces to the tensegrity structure, from which the selected cable has been omitted (damaged structure). Next, the nonlinear equation of motion of the tensegrity bridge subjected to dynamic loads is discretized and integrated in time using the unconditionally stable Newmark constant-average acceleration method combined with a Newton-Raphson iterative scheme. The dynamic simulation is initiated by cancelling the vector of external forces representing the damaged cable. For each case, the largest tension force in the cables, the largest compression force in the struts as well as the largest average midspan displacement are determined. The maximum tension obtained in all the bridge cables was way below their tension capacities for the unloaded bridge and exceeded them for only one case of the loaded one. However, the maximum compression forces obtained in the struts of the bridge were below their compression capacities. The limit deflection has been exceeded only for of the loaded bridge and for several cases of cable rupture. Nonlinear dynamic instabilities caused by cable slackening were observed in all simulations.

Dynamic snap-through of shallow spherical shells subjected to thermal shock

Present study deals with the dynamic snap-through phenomenon of an isotropic shallow spherical cap under transient type of thermal loading. The inner surface of the shell is subjected to sudden temperature elevation, whereas the outer surface is kept at reference temperature. Transient thermal shock is applied uniformly and shell thickness is assumed to be thin enough. Therefore, transient heat conduction equation is analytically solved across the thickness direction (one dimensional). Immovable simply-supported boundary conditions are assumed for the shell. Since the boundary conditions and the applied thermal shock are axisymmetric, the governing motion equations of the shell are restricted to the case of axisymmetric. First order shear deformation theory of shells is utilized to approximate the displacement field. The von Kármán type of geometrical non-linearity is used in strain-displacement relations. With the establishment of the associated Hamilton principle, differential equations of motion are extracted. Motion equations are discretized within the shell domain by means of the harmonic differential quadrature method (HDQM). The Newmark time marching scheme based on the constant average acceleration method is applied to turn the motion differential equations into a system of algebraic equations at each time step. Highly coupled equations are solved implementing the well-known Newton-Raphson iterative technique. By means of the Budiansky criterion, critical thermal shock parameters are distinguished. Critical dynamic snap-through temperatures are also verified using the phase-plane presentation. Comparison study is performed to validate the formulation and solution method of the present research with simple cases. Also, parametric investigations are performed to demonstrate the geometrical effects on the dynamic critical buckling temperatures of the shallow spherical shells subjected to thermal shock.

Nonlinear dynamic analysis of hysteretic mechanical systems by combining a novel rate-independent model and an explicit time integration method

This paper presents a computational strategy that combines a novel rate-independent phenomenological model with an explicit time integration method to efficiently perform nonlinear dynamic analyses of non-stiffening hysteretic mechanical systems. The novel rate-independent model, developed by specializing a general class of uniaxial phenomenological models, has an algebraic nature, is based on a set of only three parameters having a clear mechanical significance, and can be easily implemented in a computer program. The adopted explicit structure-dependent time integration method, belonging to the Chang’s family of explicit methods, is unconditionally stable for all non-stiffening hysteretic mechanical systems, has a second-order accuracy, does not suffer from numerical damping, and displays a small relative period error for small time step. Furthermore, it does not require iterative procedures and, consequently, does not suffer from convergence issues. Numerical accuracy and computational efficiency of the proposed procedure are assessed by performing several nonlinear time history analyses on hysteretic mechanical systems and comparing the results with those obtained by employing a conventional strategy based on the celebrated Bouc–Wen model, or its modified version, and the widely used Newmark’s constant average acceleration method.

Study on the Effect of Cable Restrainers Stiffness on the Seismic Performance of Curved Viaducts Combined use of Steel Stoppers

This paper presents analysis to evaluate the efficiency of cable restrainers (CRs) stiffness on the seismic performance of curved viaducts combined use of steel stoppers at spatial multipoint. Based on the modified Hertz-damp model, considering the energy consumption the pounding phenomena at steel stoppers was exactly simulated. Considering the elasto-plastic constitutive model and shear stiffness, CRs were also modeled. The time history analysis method is used in this analysis, considering the geometrical and material non-linearities. The uniaxial yield criteria and kinematic strain hardening criteria are adopted. The constant average acceleration method is used for the dynamic response analysis. The results show that CRs combined with steel stoppers could mitigate the pounding forces at steel stoppers, displacements of the superstructures and the curvatures at pier bases by reducing the vibration difference between superstructures. The application of CRs with high stiffness can mitigate viaduct damage in earthquakes.

A parametric analysis of the influence of the internal slug flow on the dynamic response of flexible marine risers

This paper presents a two-dimensional parametric analysis of the influence of an internal slug flow on the structural dynamic response of flexible risers. The parametric analysis allow us to study the behaviour of the parameters involved in the dynamic response of flexible risers with an internal slug flow, such as displacements, top tension, and bending moment, when any of the parameters characterizing the slug flow is modified. To accomplish this goal, a computational tool is developed. The spatial discretization of the riser is performed using beam finite elements and the corotational approach is used for the two-dimensional formulation of the beam element. The beam elements have a linear elastic behaviour and they are submitted to large displacements, although with small deformations. The incremental-iterative Newton-Raphson scheme for follower loads is used to solve the equilibrium equations in dynamic analysis. Time domain numerical integration is performed using the constant average acceleration method. The slug flow is viewed as a single-phase fluid with variable density, which is modelled by a periodic function. The numerical results show good coherence when compared to the experimental ones, qualifying the computational tool to conduct parametric studies about dynamic of flexible marine risers with an internal slug flow.

A mixed explicit–implicit time integration approach for nonlinear analysis of base-isolated structures

The paper investigates the accuracy, the stability and the computational efficiency of a mixed explicit–implicit time integration approach proposed for predicting the nonlinear response of base-isolated structures subjected to earthquake excitation. Adopting the central difference method for evaluating the response of the nonlinear base isolation system and the Newmark’s constant average acceleration method for estimating the superstructure linear response, the proposed partitioned solution approach is used to analyze a 3D seismically isolated structure subjected to a bidirectional earthquake excitation. Both isolation systems adopting lead rubber and friction pendulum bearings are considered. Numerical results show that the computational time required by the proposed method, in spite of its conditional stability arising from the use of the central difference method in the explicit integration substep, is clearly reduced in comparison to the widely used implicit time integration method adopted in conjunction with the pseudo-force approach (i.e., pseudo-force method). As a matter of fact, the typical low stiffness of the isolation system leads to a critical time step larger than the one used to define the ground acceleration accurately and the proposed method preserves its computational efficiency even in the case of isolators with very high initial stiffness (i.e., friction pendulum bearings) for which the critical time step size could become smaller.

Unified Space–Time Finite Element Methods for Dissipative Continua Dynamics

Based upon the extended framework of Hamilton’s principle, unified space–time finite element methods for viscoelastic and viscoplastic continuum dynamics are presented, respectively. For numerical efficiency, mixed time-step algorithm in time- and displacement-based algorithm in space are adopted. Through analytical investigation, we demonstrate that the Newmark’s constant average acceleration method and the present method are the same for viscoelasticity. With spatial eight-node brick elements, some numerical simulations are undertaken to validate and investigate the performance of the present non-iterative space–time finite element method for viscoplasticity.

A stabilized central difference scheme for dynamic analysis

In this work, a new, unconditionally stable, time marching procedure for dynamic analyses is presented. The scheme is derived from the standard central difference approximation, with stabilization being provided by a consistent perturbation of the original problem. Because the method only involves constitutive variables that are already available from computations at previous time steps, iterative procedures are not required to establish equilibrium when nonlinear models are focused, allowing more efficient analyses to be obtained. The theoretical properties of the proposed scheme are discussed taking into account standard stability and accuracy analyses, indicating the excellent performance of the new technique. At the end of the contribution, representative nonlinear numerical examples are studied, further illustrating the effectiveness of the new technique. Numerical results obtained by the standard central difference procedure and the implicit constant average acceleration method are also presented along the text for comparison. Copyright © 2015 John Wiley & Sons, Ltd.

A family of noniterative integration methods with desired numerical dissipation

ABSTRACTA new family of unconditionally stable integration methods for structural dynamics has been developed, which possesses the favorable numerical dissipation properties that can be continuously controlled. In particular, it can have zero damping. This numerical damping is helpful to suppress or even eliminate the spurious participation of high frequency modes, whereas the low frequency modes are almost unaffected. The most important improvement of this family method is that it involves no nonlinear iterations for each time step, and thus it is very computationally efficient when compared with a general second‐order accurate integration method, such as the constant average acceleration method. Copyright © 2014 John Wiley & Sons, Ltd.

Dynamic analysis of delaminated composite conical shells under low velocity impact

The dynamic response of graphite-epoxy delaminated cross-ply composite shallow conical shells subjected to low velocity normal impact is investigated using the finite element method. An eight-noded isoparametric plate bending element is employed in the formulation incorporating rotary inertia and effects of transverse shear deformation based on Mindlin’s theory. In the time integration, Newmark constant average acceleration method in conjunction with an efficient iteration scheme is used. The modified Hertzian contact law which accounts for permanent indentation is utilized to compute the time-dependent equations. Numerical results present the histories of contact force, deflection/thickness and impactor’s displacement for centrally impacted delaminated cross-ply composite cantilevered conical shells.

A Taylor Series Integration Method with Coupling in Structural Dynamics

Based on the coupled precise time integration method and basic assumptions of constant average acceleration method in Newmark family, implicit series solution of structural dynamic equation is put forward by introducing the Taylor series expansion. Relevant time step integration formulas were designed. Stability and accuracy of the method were analyzed. Stability analyses show that the coupling implicit method is stable when damping ratio is equal to 0, and is conditionally stable when damping ratio are other values. The results show that the accuracy of the algorithm can be controlled by choosing the number of truncation order of Taylor series expansion and is better than that of traditional scheme with the increase of time step. Number examples are given to demonstrate the validity of the proposed method.

Behavior of a viscoelastic composite plates under transient load

An efficient numerical algorithm for analyzing dynamic response of orthotropic viscoelastic composite laminates has been developed in the time domain. The integral form of the constitutive laws is exploited. Generalized Wiechert model is adopted to simulate the viscoelasticity of the structure Mindlin-Reissner plate theory is utilized in finite element formulation employing the consistent mass matrix. The proposed algorithm is incorporated in the framework of the finite element based on the displacement model. Newmark Constant average acceleration method is used for integration equation of motion. The developed recurrence formula permits the new time solutions to be evaluated using only previous time values. The developed solution technique is applied to orthotropic plate under two types of force, which are the step-pulse and sin-pulse force.