Pseudo-force Method Research Articles

Nonlinear vibrations of beams with fractional derivative elements crossed by moving loads

This paper addresses the evaluation of the time-domain response of nonlinear beams endowed with a fractional derivative element crossed by moving loads. Nonlinearities originate from the assumption of moderately large displacements of the beam. Following a Galerkin-type solution procedure, beam transversal displacement is represented in terms of the linear modes of vibration and time-dependent generalized coordinates. A novel step-by-step integration scheme, labeled improved pseudo-force method (IPFM), is developed for the numerical solution of the set of coupled nonlinear fractional differential equations governing the time-dependent generalized coordinates. The proposed procedure stems from the extension of a recently developed step-by-step scheme for the dynamic analysis of fractional single-degree-of-freedom systems. The IPFM involves the following main steps: i) to apply the Grünwald–Letnikov approximation of the fractional derivative; ii) to treat terms depending on the unknown values of the response as pseudo-forces; iii) to handle nonlinearities by performing iterations at each time step.Numerical results are presented to assess the accuracy of the IPFM as well as to investigate the influence of the fractional derivative order and coefficient on nonlinear beam vibrations under moving loads.

Improved pseudo-force approach for Monte Carlo Simulation of non-linear fractional oscillators under stochastic excitation

This paper presents a step-by-step procedure for the numerical integration of the fractional differential equation governing the response of single-degree-of-freedom (SDOF) non-linear systems endowed with fractional derivatives subjected to stochastic excitation. The procedure, labeled improved pseudo-force method (IPFM), is developed by extending a step-by-step integration scheme proposed by the second author for the numerical solution of classical differential equations. The IPFM relies on the following main steps: i) to use the Grünwald–Letnikov (GL) approximation of the fractional derivative; ii) to treat terms depending on the unknown values of the response, which result from the GL approximation as well as from the non-linear restoring forces, as pseudo-forces; iii) to handle non-linearities by performing iterations at each time step. The IPFM provides accurate solutions by using time steps of larger size compared to classical step-by-step integration schemes. In this paper, the IPFM is applied within the framework of classical Monte Carlo Simulation (MCS) to evaluate the time domain dynamic response of non-linear fractional systems subjected to the generic sample of a stochastic excitation.

Falling-Film Absorption Model Considering Surface Wave and Vibration Effects Based on Lattice Boltzmann Method

As one of the effective solutions to recover waste heat, absorption refrigeration systems are used in various industrial or refrigeration places. Flat-plate falling-film absorption is one of the newer types, and the lattice Boltzmann method (LBM) has certain advantages compared with the traditional numerical simulation method. In this work, an LBM is used to analyze flat-plate falling-film absorption. Using the additional calculation of the pressure by the pseudo-force model, a lithium bromide–water working fluid–heat and mass transfer model driven by steam partial pressure is realized. The results show that the turbulence generated in the surface wave has a favorable effect on the absorption process; the degree of turbulence gradually decreases with the increase in the Reynolds number, which weakens the increasing effect of the surface wave on the absorption. When the Reynolds number is moderate, the solitary wave flows forward relative to the front thin liquid film, which promotes concentration and temperature diffusion inside the liquid film and inside the solitary wave. The model of falling-film flow under vibration environment is realized by using the characteristic of imposing inertial force in the model by pseudo-force method. The results show that vibration has a favorable effect on liquid film absorption, increasing the amplitude can increase the gas–liquid contact area and obtain a lower average film thickness, while increasing the vibration frequency can promote the internal diffusion of the solution.

Transient elastodynamic analysis with a combination of convolution quadrature method and pseudo-initial condition method

Purpose The Convolution Quadrature Method (CQM) has been widely applied to solve transient elastodynamic problems because of its stability and generality. However, the CQM suffers from the problems of huge memory requirement in case of direct implementation in time domain or CPU time in case of its reformulation in Laplace domain. The purpose of this paper is to combine the CQM with the pseudo-initial condition method (PICM) to achieve a good balance between memory requirement and CPU time. Design/methodology/approach The combined methods first subdivide the whole analysis into a few sub-analyses, which is dealt with the PICM, namely, the results obtained by previous sub-analysis are used as the initial conditions for the next sub-analysis. In each sub-analysis, the time interval is further discretized into a number of sub-steps and dealt with the CQM. For non-zero initial conditions, the pseudo-force method is used to transform them into equivalent body forces. The boundary face method is employed in the numerical implementation. Three examples are analyzed. Results are compared with analytical solutions or FEM results and the results of reformulated CQM. Findings Results demonstrate that the computation time and the storage requirement can be reduced significantly as compared to the CQM, by using the combined approach. Originality/value The combined methods can be successfully applied to the problems of long-time dynamic response, which requires a large amount of computer memory when CQM is applied, while preserving the CQM stability. If the number of time steps is high, then the accuracy of the proposed approach can be deteriorated because of the pseudo-force method.

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.

Inelastic Seismic Response of Building with Friction Damper

Dampers are used to control response of structure in recent years. Seismic response is reduced by adding energy dissipation capacity to the building. Dampers are provided to dissipate energy to reduce the damages on buildings. Relative displacement of two floors are opposed by damper elements. Dampers dissipate energy with no or little degradation in strength and stiffness. This work presents the response of structure with energy dissipation device i.e. friction damper built on soft soil when the structure is subjected to a ground motion. Parameters that influence strength and stiffness of structure and energy dissipating device are studied. Response of structure with energy dissipating device is analyzed and it is shown that building with damper shows better performance during earthquake. Building structures are designed using results from elastic analysis, though inelastic behavior well observed during the earthquake. Actual response of the structure can be estimated in the inelastic range by using nonlinear structural analysis programs. Inelastic Damage indices of structural component, overall building are computed by formulating Modified Park and Ang model. Nonlinear analysis program IDARC 2D is used for modeling and analysis. Modeling of friction damper is done using a Wen–Bouc model without stiffness and strength or strength degradation. These devices are modeled with an axial diagonal element. Pseudo force approach is introduced to consider the forces in damper elements, that is, forces in damper elements are subtracted from external load vector. Smooth hysteretic model is used to model response of friction dampers. Nonlinear dynamic analysis is carried out using Newmark-Beta integration method and the pseudo-force method. Damaged state of structure is obtained to observe whether elements are cracked, yielded or failed during analysis. Response parameters such as lateral floor displacement, storey drift, base shear are also computed.

Closed-form criterion for convergence and stability of pseudo-force method for nonlinear dynamic analysis

The pseudo-force (PF) method has been successfully applied to numerous problems with positive secant stiffness or positive tangential stiffness; however, it is widely believed to diverge for problems with the ratio of actual stiffness to pseudo-tangential stiffness being negative or larger than two. In the present paper, this point of view is investigated by applying PF method to single- and multi-degree of freedom systems with elasto-plastic nonlinearities which involve repeated cycles of loading and unloading, where the elastic behavior is either very high positive tangential stiffness or very high negative tangential stiffness (e.g., in Coulomb friction or velocity-dependent friction) or geometrically nonlinear problems. The objective of this study is to in essence seek the reasons for convergence of PF method in the presence of stiff nonlinearities. Closed-form expressions for convergence and stability criteria are derived. Approximate solution by modified Lindstedt–Poincare method (L–P method) is formulated. The PF method is also applied to a base-isolated structure with nonlinear sliding or velocity-dependent friction bearings. The results of the study indicate the wide applicability of PF method for highly nonlinear problems, including those with high positive or negative tangential stiffness.

Extraction of free body frequency response functions from flexibly constrained test

The traditional way of low stiffness suspension may no longer work for very large scale structures with extremely low first flexible natural frequency. In this paper, we replace the low stiffness suspension with a group of supports. After separate Frequency Response Function tests of the whole supported structure and the support structures, the influence of the supports could be removed from a later computation. To this end, the identification method and support design are critical. The existing substructure decoupling methods are briefly reviewed and some incomplete issues in the earlier works are discussed. Then the relationship among the methods is investigated. With these studies, some improvements on the pseudo-force method are made. The proper design of supports and measurement locations are also studied. Both numerical and experimental examples are described in the paper.

Fast stochastic analysis for non-proportionally damped system

Fundamental principles from structural dynamics, pseudo excitation method and pseudo force method are used to develop a new fast stochastic method for seismic response analysis of non-proportionally damped system. In the method, the mathematical equation of non-proportionally damped system is expressed in the iterative form, based on which the inverse operation of the matrices is avoided. Moreover, the new method does not need the solution of any complex eigenvalue problem, in contrast to other methods found in the literature.

Micromechanics model for static and dynamic strength of concrete under confinement

The process of propagation, kinking of microcracks in concrete and the interaction among cracks as well as the induced failure were analyzed using the model that describes the wing type crack from the point of view of micromechanics. The pseudo-force method is applied to calculate the compressive strength factor of kinky propagated crack taking into account the effect of interaction among cracks. On the assumption that the micro fracture toughness of concrete does not vary with stain rate, the static and dynamic strength of concrete under different confinements can be calculated. The comparison of calculation result with experimental data indicates that a good agreement is achieved which implies that the model can be used to explain the rate-dependent properties of concrete in multi-axial stress state.

Contact Interaction Between a Laminated Shell of Revolution and a Rigid or Elastic Foundation

A method is proposed to solve the contact problem for laminated anisotropic shells of revolution. The method is based on a two-dimensional model that accounts for transverse shears and reduction. Also the method is based on the method of successive approximations, the generalized pseudo-force method, and a numerical-analytical method of solving boundary-value problems. The results obtained for a cylindrical shell of complex thickness structure are compared with those obtained in three-dimensional formulation

An efficient time/frequency domain algorithm for modal analysis of non-linear models discretized by the FEM

The present paper describes an efficient time/frequency domain approach for modal analysis of non-linear models discretized by the FEM. An efficient recurrence relationship for displacement and velocity, based on the Green’s function of the model and its time derivative, is presented in both a time and a frequency domain context. In the time domain procedure the Green’s function related terms are analytical; in the frequency domain procedure the fundamental terms are indirectly evaluated, taking into account steady-state responses of the system. The pseudo-force method is adopted in order to deal with the model non-linearity. Results for the time/frequency domain approaches discussed here showed the same level of accuracy as those obtained by the Newmark/Modified-Newton algorithm at a lower computational cost.

A pseudo‐force iterative method with separate scale factors for dynamic analysis of structures with non‐proportional damping

AbstractDynamic equilibrium equations of structural systems with non‐proportional damping are coupled through the damping terms. Such coupling invalidates application of the classical modal superposition method. In this paper, a mode‐superposition pseudo‐force method is proposed. The coupled equilibrium equations are solved by an iterative process in which the coupling terms are treated as pseudo‐forces. A scale factor for each mode of the system is obtained by optimizing the iteration convergence. Through these uniquely solved scale factors, the modified modal equations not only converge much faster but also yield results with higher accuracy. A proof of the convergence of the iterative process is also presented. Copyright © 2002 John Wiley & Sons, Ltd.

Pseudoforce Method for Nonlinear Analysis and Reanalysis of Structural Systems

This paper develops a new solver to enhance the computational efficiency of finite-element programs for the nonlinear analysis and reanalysis of structural systems. The proposed solver does not require the reassembly of the global stiffness matrix and can be easily implemented in present-day finite-element packages. It is particularly well suited to those situations where a limited number of members are changed at each step of an iterative optimization algorithm or reliability analysis. It is also applicable to a nonlinear analysis where the plastic zone spreads throughout the structure due to incremental loading. This solver is based on an extension of the Sherman-Morrison-Woodburg formula and is applicable to a variety of structural systems including 2D and 3D trusses, frames, grids, plates, and shells. The solver defines the response of the modified structure as the difference between the response of the original structure to a set of applied loads and the response of the original structure to a set of pseudoforces. The proposed algorithm requires O(mn) operations, as compared with traditional solvers that need O(m2n) operations, where m = bandwidth of the global stiffness matrix and n = number of degrees of freedom. Thus, the pseudoforce method provides a dramatic improvement of computational efficiency for structural redesign and optimization problems, since it can perform a nonlinear incremental analysis no harder than the inversion of the global stiffness matrix. The proposed method's efficiency and accuracy are demonstrated in this paper through the nonlinear analysis of an example bridge and a frame redesign problem.

Analysis of building frames with viscoelastic dampers under base excitation

A frequency domain response analysis is presented for building frames passively controlled by viscoelastic dampers, under harmonic ground excitation. Three different models are used to represent the linear dynamic force-deformation characteristics of viscoelastic dampers namely, Kelvin model, Linear hysteretic model and Maxwell model. The frequency domain solution is obtained by (i) an iterative pseudo-force method, which uses undamped mode shapes and frequencies of the system, (ii) an approximate modal strain energy method, which uses an equivalent modal damping of the system in each mode of vibration, and (iii) an exact method which uses complex frequency response function of the system. The responses obtained by three different methods are compared for different combinations of viscoelastic dampers giving rise to both classically and non-classically damped cases. In addition, the effect of the modelling of viscoelastic dampers on the response is investigated for a certain frequency range of interest. The results of the study are useful in appropriate modelling of viscoelastic dampers and in understanding the implication of using modal analysis procedure for building frames which are passively controlled by viscoelastic dampers against base excitation.

A HYBRID PSEUDO-FORCE/LAPLACE TRANSFORM METHOD FOR NON-LINEAR TRANSIENT RESPONSE OF A SUSPENDED CABLE

A hybrid numerical scheme involving the combination of the Laplace transform technique and the pseudo-force method is proposed to analyze the non-linear transient response of a suspended cable subjected to arbitrary dynamic loading. A theoretical model of the cable with multi-degree-of-freedom is first obtained through discretization of the partial differential equations by finite difference approximation. The non-linear governing equations take into account the effects of quadratic and cubic geometric non-linearities. The proposed method deals with the non-linear effects as pseudo-forces and then establishes an iterative solution scheme in the alternating Laplace/time domain by means of fast numerical Laplace transform. This method eschews a time-stepping process and therefore is computationally efficient. It also readily deals with the viscoelastic damping with frequency-dependent model parameters and the hysteresis damping in terms of complex stiffness models. Numerical examples are presented to evaluate the dynamic responses of suspended cables under a concentrated sinusoidal force and a distributed random excitation, and to identify the non-linear response properties by comparison with the linear vibration. The validity and accuracy of the proposed method is also verified by comparing the results with those obtained by using the direct time integration.

A frequency-domain pseudo-force method for dynamic structural analysis nonlinear systems and nonproportional damping

A frequency-domain method for nonlinear analysis of structural systems with viscous, hysteretic, nonproportional and frequency-dependent damping is presented. The nonlinear effects and nonproportional damping are considered through pseudo-force terms. The modal coordinates uncoupled equations are iteratively solved. The treatment of initial conditions in the frequency domain which is necessary for the treatment of the uncoupled equations is initially adressed.

Non-linear transient heat transfer analysis based on Lanczos co-ordinates using pseudo-force method

This paper describes a reduced finite element formulation for mildly non-linear transient heat transfer analysis problems based on the Lanczos algorithm. In the proposed reduced formulation, all material non-linearities of irradiation boundary elements are included using the pseudo-force method. Numerical time integration of the reduced formulation is conducted by the Galerkin method. The results of numerical examples demonstrate the applicability and the accuracy of the proposed method for transient heat transfer analysis when restricted to mildly non-linear situations. In particular, the example of a bridge under environmental thermal condition shows that the proposed method is about 20 times faster, without loss of solution stability and accuracy, than a direct integration method for the unreduced formulation. Copyright © 1999 John Wiley & Sons, Ltd.

A modal procedure for seismic analysis of non-linear base-isolated multistorey structures

A modal procedure for non-linear analysis of multistorey structures with high-damping base-isolation systems was proposed. Two different isolation devices were considered in the analysis: an high-damping laminated rubber bearing and a lead-rubber bearing. Starting from deformational properties verified by tests, the isolation systems were characterized using three different analytical models (an Elastic Viscous, a Bilinear Hysteretic and a Wen's Model) with parameters depending from maximum lateral strain. After non-linear modelling of isolation and lateral-force-resisting systems, the effects of material non-linearities were considered as pseudo-forces applied to the equivalent linear system (Pseudo-Force Method) and the formally linearized equations of motion were uncoupled by the transformation defined by the complex mode shapes. The modal responses were finally obtained with an extension of Nigam–Jennings technique to non-linear and non-classically damped systems, in conjunction with an iterative technique searching for non-linear contributions satisfying equations of motion and constitutive laws. Since the properties of the isolated structure usually change with maximun lateral strain of isolation bearings, the integration of a new set of governing equations was required for each design-displacement value. The procedure proposed was described in detail and then applied for the determination of modal and total seismic responses in some real cases. At first, a very good agreement between non-linear responses obtained with the proposed mode superposition and with a direct integration method was observed. Then a comparison of results obtained with the three different analytical models of the isolation bearings was carried out. At last, the exact modal response obtained with analytical models depending from the design displacement of the isolation bearings was compared with two different approximated solutions, evaluated using mode shapes and isolation properties, respectively, calculated under simplified hypothesis.© 1998 John Wiley & Sons, Ltd.

Higher order statistics of the response of MDOF linear systems under polynomials of filtered normal white noises

This paper exploits the work presented in the companion paper in order to evaluate the higher order statistics of the response of linear systems excited by polynomials of filtered normal processes. In fact, by means of a variable transformation, the original system is replaced by a linear one excited by external and linearly parametric white noise excitations. The transition matrix of the new enlarged system is obtained simply once the transition matrices of the original system and of the filter are evaluated. The method is then applied in order to evaluate the higher order statistics of the approximate response of nonlinear systems to which the pseudo-force method is applied.