Lumped-parameter model and recursive evaluation of interaction forces of semi-infinite uniform fluid channel for time-domain dam-reservoir analysis

To calculate the hydrodynamic interaction forces of the reservoir directly in the time-domain, the dynamic stiffness of each mode of the semi-infinite uniform fluid channel is either represented by a lumped-parameter model with frequency-independent real coefficients of the springs, dashpots and masses and with only a few additional internal degrees of freedom, or the interaction forces are calculated recursively. For each mode characterized by its eigenvalue, the coefficients of the lumped-parameter model and the recursive coefficients are specified, which can be used directly in a practical application. The procedures exhibit many advantages: the only approximation (replacing the rigorous dynamic stiffness by a ratio of two polynomials) can be evaluated visibly. No unfamiliar discrete-time manipulations such as the z-transformation are used. The stiffness, damping and mass matrices corresponding to the lumped-parameter model are automatically symmetrical. Stability of the procedures is also guaranteed. Combining the lumped-parameter model of the semi-infinite uniform channel with the finite-element discretization of the irregular fluid region or calculating the interaction forces recursively allows a reservoir of arbitrary shape to be analysed directly in the time domain. Non-linearities in the dam can, thus, be taken into consideration in a seismic analysis.     

Consistent lumped-parameter models for unbounded soil: Physical representation

A systematic procedure to develop a consistent lumped-parameter model with real frequency-independent coefficients to represent the unbounded soil is developed. Each (modelled) dynamic-stiffness coefficient in the frequency domain is approximated as a ratio of two polynomials, which is then formulated as a partial-fraction expansion. Each of these terms is represented by a discrete model, which is the building block of the lumped-parameter model. A second-order term, for example, leads to a discrete model with springs and dampers with two internal degrees of freedom, corresponding to two first-order differential equations, or, alternatively, results in a discrete model with springs, dampers and a mass with one internal degree of freedom, corresponding to one second-order differential equation. The lumped-parameter model can easily be incorporated in a general-purpose structural dynamics program working in the time domain, whereby the structure can even be non-linear. A thorough evaluation shows that highly accurate results are achieved, even for dynamic systems with a cutoff frequency.     

PROPERTY MATRICES IDENTIFICATION OF UNBOUNDED MEDIUM FROM UNIT-IMPULSE RESPONSE FUNCTIONS USING LEGENDRE POLYNOMIALS: FORMULATION

A systematic procedure to construct the (symmetric) static-stiffness, damping and mass matrices representing the unbounded medium is presented addressing the unit-impulse response matrix corresponding to the degrees of freedom on the structure–medium interface. The unit-impulse response matrix is first diagonalized which then permits each term to be modelled independently from the others using expansions in a series of Legendre polynomials in the time domain. This leads to a rational approximation in the frequency domain of the dynamic-stiffness coefficient. Using a lumped-parameter model which provides physical insight the property matrices are constructed.     

Verification of system identification utilizing shaking table tests of a full‐scale 4‐story steel building

This paper verifies the feasibility of the proposed system identification methods by utilizing shaking table tests of a full‐scale four‐story steel building at E‐Defense in Japan. The natural frequencies, damping ratios and modal shapes are evaluated by single‐input‐four‐output ARX models. These modal parameters are prepared to identify the mass, damping and stiffness matrices when the objective structure is modelled as a four degrees of freedom (4DOF) linear shear building in each horizontal direction. The nonlinearity in stiffness is expressed as a Bouc–Wen hysteretic system when it is modelled as a 4DOF nonlinear shear building. The identified hysteretic curves of all stories are compared to the corresponding experimental results. The simple damage detection is implemented using single‐input‐single‐output ARX models, which require only two measurements in each horizontal direction. The modal parameters are equivalent‐linearly evaluated by the recursive Least Squares Method with a forgetting factor. When the structure is damaged, its natural frequencies decrease, and the corresponding damping ratios increase. The fluctuation of the identified modal properties is the indirect information for damage detection of the structure. Copyright © 2015 John Wiley & Sons, Ltd.     

System identification of linear structures based on Hilbert–Huang spectral analysis. Part 1: normal modes

Based on the Hilbert–Huang spectral analysis, a method is proposed to identify multi‐degree‐of‐freedom (MDOF) linear systems using measured free vibration time histories. For MDOF systems, the normal modes have been assumed to exist. In this method, the measured response data, which are polluted by noises, are first decomposed into modal responses using the empirical mode decomposition (EMD) approach with intermittency criteria. Then, the Hilbert transform is applied to each modal response to obtain the instantaneous amplitude and phase angle time histories. A linear least‐square fit procedure is proposed to identify the natural frequency and damping ratio from the instantaneous amplitude and phase angle for each modal response. Based on a single measurement of the free vibration time history at one appropriate location, natural frequencies and damping ratios can be identified. When the responses at all degrees of freedom are measured, the mode shapes and the physical mass, damping and stiffness matrices of the structure can be determined. The applications of the proposed method are illustrated using three linear systems with different dynamic characteristics. Numerical simulation results demonstrate that the proposed system identification method yields quite accurate results, and it offers a new and effective tool for the system identification of linear structures in which normal modes exist. Copyright © 2003 John Wiley & Sons, Ltd.     

Modal equations of linear structures with viscoelastic dampers

Several types of energy dissipation devices using viscoelastic materials have been proposed to reduce vibration in structures subjected to wind and earthquake excitations. At constant temperature and small strain levels, the mechanical behaviour of Viscoelastic (VE) materials can be described using linear operators. In general, the stiffness and damping matrices of structures using VE devices are frequency dependent; this implies that the classical second-order differential equations for the modal co-ordinates are not a complete model for this type of structures. In this paper, the concept of modal coupling in the frequency domain is addressed, expressions for diagonalizable frequency-dependent stiffness and damping matrices are given, and an iterative technique for the computation of the response of viscoelastic structures is studied. Necessary and sufficient conditions for convergence of the technique are given and numerical examples are developed to illustrate the application of the method.     

Estimated mass and stiffness matrices of shear building from modal test data

A method that estimates mass and stiffness matrices of shear building from modal test data is presented in this paper. The method depends on only measurable points that are less in number than the total structural degrees of freedom, and on the first two orders of structural mode measured. So it is applicable to most of the general test. Based on this method modal data of unmeasurable points are estimated, then global mass and stiffness matrices of structure are obtained by using the first two orders of modal data. Taking advantage of iteration the optimum global mass and stiffness matrices are gained. Finally, an example is studied in this paper. Its result shows that this method is reliable. © 1998 John Wiley & Sons, Ltd.     

Identification of structural parameters of multistorey shear buildings from modal data

A procedure has been presented in this paper to identify the structural parameters, viz. mass and stiffness matrices, from modal test data for multistorey shear buildings. The first two orders of modal data have been used by other researchers to estimate the global matrices where they depend only on measurable points which are less than the total number of structural degrees of freedom. The above method has been refined here by using Holzer criteria along with other numerical methods to estimate the global mass and stiffness matrices of the structure. This shows the methodology to be more efficient and accurate. The reliability of the procedure has been shown by examples of multistorey buildings. Copyright © 2004 John Wiley & Sons, Ltd.     

Global lumped-parameter model with physical representation for unbounded medium

A systematic procedure to construct a consistent global lumped-parameter model consisting of springs, dashpots and possibly masses with frequency-independent coefficients connecting the degrees of freedom of the nodes of any structure-medium interface for the unbounded medium is presented. The dynamic-stiffness matrix is first diagonalized which then permits each term to be modelled independently from the others. Physical insight is thus provided. Alternatively, the (symmetric) static-stiffness and damping matrices and possibly mass matrix of the unbounded medium can be established directly.     

Numerical artifacts associated with Rayleigh and modal damping models of inelastic structures with massless coordinates

The paper presents a detailed reexamination of the effects of three damping models on the inelastic seismic response of structures with massless degrees of freedom. The models considered correspond to (a) Rayleigh damping based on current properties (tangent stiffness), (b) Rayleigh damping based on initial properties, and (c) modal damping. The results suggest that some nonzero damping forces/moments at massless DOFs obtained in multistory frames for the case of Rayleigh damping with tangent stiffness may be numerical artifacts rather than a deficiency of the damping model. The results also indicate that significant artificial numerical oscillations in the velocities of the secondary components of MDOF structures are introduced when modal damping or mass-proportional damping is used.     

Implementation and application of the unconditionally stable explicit parametrically dissipative KR‐α method for real‐time hybrid simulation

In real‐time hybrid simulations (RTHS) that utilize explicit integration algorithms, the inherent damping in the analytical substructure is generally defined using mass and initial stiffness proportional damping. This type of damping model is known to produce inaccurate results when the structure undergoes significant inelastic deformations. To alleviate the problem, a form of a nonproportional damping model often used in numerical simulations involving implicit integration algorithms can be considered. This type of damping model, however, when used with explicit integration algorithms can require a small time step to achieve the desired accuracy in an RTHS involving a structure with a large number of degrees of freedom. Restrictions on the minimum time step exist in an RTHS that are associated with the computational demand. Integrating the equations of motion for an RTHS with too large of a time step can result in spurious high‐frequency oscillations in the member forces for elements of the structural model that undergo inelastic deformations. The problem is circumvented by introducing the parametrically controllable numerical energy dissipation available in the recently developed unconditionally stable explicit KR‐α method. This paper reviews the formulation of the KR‐α method and presents an efficient implementation for RTHS. Using the method, RTHS of a three‐story 0.6‐scale prototype steel building with nonlinear elastomeric dampers are conducted with a ground motion scaled to the design basis and maximum considered earthquake hazard levels. The results show that controllable numerical energy dissipation can significantly eliminate spurious participation of higher modes and produce exceptional RTHS results. Copyright © 2014 John Wiley & Sons, Ltd.     

Evaluating foundation mass,damping and stiffness by the least‐squares method

This paper discusses how to use the three‐dimensional (3D) time‐domain finite‐element method incorporating the least‐squares method to calculate the equivalent foundation mass, damping and stiffness matrices. Numerical simulations indicate that the accuracy of these equivalent matrices is acceptable when the applied harmonic force of 1+sine is used. Moreover, the accuracy of the least‐squares method using the 1+sine force is not sensitive to the first time step for inclusion of data. Since the finite‐element method can model problems flexibly, the equivalent mass, damping and stiffness matrices of very complicated soil profiles and foundations can be established without difficulty using this least‐squares method. Copyright © 2003 John Wiley & Sons, Ltd.     

Curved beam finite elements for coupled bending and torsional vibration

Curved beam finite elements are presented for out of plane coupled bending and torsional vibration. The element formulation is based upon the exact differential equations of an infinitesimal element in static equilibrium. The effects of shear deformation and rotary inertia are allowed for in the analysis. The element stiffness and mass matrices can be easily restricted to those of a ‘thin’ beam without the secondary effects. Frequencies obtained using either formulation are shown to converge onto exact values using ‘thick’ or ‘thin’ beam theories.     

Representation of radiation damping by fractional time derivatives

When modelling unbounded domains, formulation of a matrix‐valued force–displacement relationship which can take radiation damping into account is of major importance. In this paper, a method to describe the dynamic stiffness by a system of fractional differential equations in the time‐domain is presented. Here, a doubly asymptotic rational approximation of the low‐frequency force–displacement relationship is used, whereas a direct interpretation of the asymptotic part as a fractional derivative is possible. The numerical solution of the corresponding system of fractional differential equations is demonstrated using the infinite beam on elastic foundation as an example. Copyright © 2003 John Wiley & Sons, Ltd.     

A numerical study of damping

Diagonal damping matrices were computed for three systems which have non-proportional damping matrices. These diagonal damping matrices were computed on three bases, as follows: 1. After normalizing the equations of motion by the modal matrix, the diagonal terms are retained ignoring the non-diagonal terms. 2. Diagonal damping matrix is established by the optimization algorithm which minimizes the mean square error of the frequency response. 3. Diagonal damping is determined from the normalized differential equation by matching the peaks of the coupled and uncoupled system. The frequency responses for the three cases of one of the three systems are presented together with a comparison of the energy dissipation.     

Simple formulas for the dynamic stiffness of pile groups

Simple formulas are derived for the dynamic stiffness of pile group foundations subjected to horizontal and rocking dynamic loads. The formulations are based on the construction of a general model of impedance matrices as the condensation of matrices of mass, damping, and stiffness, and on the identification of the values of these matrices on an extensive database of numerical experiments computed using coupled finite element–boundary element models. The formulations obtained can be readily used for the design of both floating piles on homogeneous half‐space and end‐bearing piles and are applicable for a wide range of mechanical and geometrical parameters of the soil and piles, in particular for large pile groups. For the seismic design of a building, the use of the simple formulas rather than a full computational model is shown to induce little error on the evaluation of the response spectra and time histories. Copyright © 2009 John Wiley & Sons, Ltd.     

Optimum viscous damper for connecting adjacent SDOF structures for harmonic and stationary white‐noise random excitations

The dynamic behaviour of two adjacent single‐degree‐of‐freedom (SDOF) structures connected with a viscous damper is studied under base acceleration. The base acceleration is modelled as harmonic excitation as well as stationary white‐noise random process. The governing equations of motion of the connected system are derived and solved for relative displacement and absolute acceleration responses of connected structures. The response of structures is found to be reduced by connecting with a viscous damper having appropriate damping. For undamped SDOF structures, the closed‐form expressions for optimum damping of viscous damper for minimum steady state as well as minimum mean square relative displacement and absolute acceleration of either of the connected SDOF structures are derived. The optimum damper damping is found to be functions of mass and frequency ratio of two connected structures. Further, numerical results had indicated that the damping of the connected structures does not have noticeable effects on the optimum damper damping and the corresponding optimized response. This implies that the derived closed‐form expressions for optimum damper damping of undamped structures can also be used in practical applications for damped structures. Copyright © 2006 John Wiley & Sons, Ltd.     

Pipeline response to random ground motion by discrete model

The response of buried pipelines to random excitation by earthquake forces is obtained using a lumped mass model. The earthquake is considered as a stationary random process characterized by a power spectral density function (PSDF). The cross spectral density function between two random inputs along the length of the pipe is defined with the help of the local earthquake PSDF which is the same for all points, and a frequency dependent exponentially decaying (with distance) function. Soil resistance to dynamic excitation along the pipelength is obtained in an approximate manner with the help of frequency independent impedance functions derived from half-space analysis and Mindlin's static stresses within the soil due to point loads. The proposed method has the advantage that it can take into consideration the cross terms in soil stiffness and damping matrices and can consider any boundary condition that needs to be satisfied at the ends of the pipe. A parametric study is also made to show the influence of cross terms in the soil stiffness and damping matrices on the response of the pipe.     

Dynamic analysis of column-supported hyperboloidal shells

The dynamic response of hyperboloidal shells on discrete column supports is studied using a curved rotational shell finite element. In this finite element formulation, the displacement field over each element domain is approximated by polynomial functions in which the coefficients of the linear terms correspond to the nodal values of the displacements and the higher order terms vanish at the nodal circles. The stiffness and mass matrices associated with the equations of motion are derived from Hamilton's variational principle and include the effects of transverse shearing deformation and rotatory inertia. Since the formulation, as such, involves a great many degrees of freedom because of the use of higher order displacement functions, the kinematic condensation technique is employed to reduce the order of the dynamic problem The dynamic analysis indicates the importance of realistically modelling the base region of the shell. Studies on a prototype tower indicates that the base flexibility reduces the natural frequencies of the shell and increases the displacements near the base. The magnitude of this reduction, which could be significant, depends primarily on the tangential stiffnesses of the supporting columns and is hardly affected by the thickened ring beam at the base.