Equivalent Linearization Technique Research Articles

Response of frictional base isolation systems to horizontal-vertical random earthquake excitations

Responses of a rigid structure with a frictional base isolation system subjected to random horizontal-vertical earthquake excitations are studied. The ground accelerations are modelled by segments of stationary and nonstationary Gaussian white noise and filtered white noise processes. The technique of nonstationary equivalent linearization is used for the response analysis. The differential equation governing the covariance matrix is solved and for a range of values of parameters, the root-mean-square displacement and velocity responses of the structure are calculated. The results are compared with those obtained by a series of Monte-Carlo digital simulations and reasonable agreement is observed. For several values of parameters, responses of the structure to the accelerograms of the North-South and vertical components of El Centro 1940 earthquake are studied. Particular attentions are given to the effects of vertical excitation and the elastic stiffness of the isolation system. It is observed that vertical excitation amplifies the horizontal slip of the structure to an extent. The amount of amplification under various conditions are evaluated and discussed. It is also shown that the slip displacement decreases sharply as the elastic restoring force of the isolator increases.

Vibration control of a cylindrical off-shore structure.

In the design of an off-shore tower-like structure, which consists of two pipe sections assembled with universal joints and attached to a buried sinker block on the sea bed, enough buoyancy can be provided to keep the structure vertical. It is desirable to reduce both the structural oscillations due to random wave excitation forces and the reaction forces acting on the sinker block. Assuming that the motion of the structure is two-dimensional in the vertical plane, these design requirements can be effectively satisfied if appropriate restoring and damping coefficients are provided at the joints. In this study, we employed the statistical equivalent linearization technique to linearize a nonlinear fluid drag force, and applied a sub-optimal control theory with control structural constraints to obtain the optima stiffness and damping coefficients required at the joints after a trade-off between the rms values of the top displacement of structure and the rms values of control moments at the joints.

Nonstationary response analysis of a nonsymmetric nonlinear multi-degree-of freedom system to nonwhite random excitation.

An approximate analytical technique is developed for obtaining the nonstationary responses of a multi-degree-of-freedom mechanical system with nonsymmetric non-linearities subjected to nonwhite random excitations with arbitrary covariance functions. The method consists of a modification of the equivalent linearization technique, in which the equivalent linear system is subjected to the equivalent forcing function with shifted mean, and the use of moment equations of the system. Numerical examples are given for single and two degree-of-freedom systems whose restoring force characteristics have both quadratic and cubic nonlinearities. In the first case, the results are compared with those of the corresponding digital simulation in order to demonstrate the validity of the present method.

Equivalence of single-term Wiener-Hermite and equivalent linearization techniques

It is shown that when only the first term of the Wiener-Hermite series is considered, the method becomes equivalent to the time-dependent equivalent linearization technique.

Fuzzy random vibration of multi‐degree‐of‐freedom hysteretic systems subjected to earthquakes

AbstractEarthquake excitation not only has evident randomness but also has strong fuzziness owing to the uncertainties in the definition of earthquake intensity and site soil classification. In this paper, the seismic ground motion is simulated as a stationary filtered white noise with fuzzy parameters, an analytical procedure is proposed to analyse the fuzzy random vibration of multi‐degree‐of‐freedom (MDOF) hysteretic systems, and the covariance matrix of the fuzzy random responses is derived by the equivalent linearization technique. Finally, some numerical results for a two‐storey shear hysteretic frame are presented to demonstrate the validity and effectiveness of the procedure proposed.

Reliability and Durability of Marine Structures

A study of the dynamic response of offshore towers to wind‐generated random ocean waves is presented. The analysis is performed in the frequency domain and the equations of motion are solved using equivalent linearization techniques. Two methods of reliability analysis are considered in this study: a statistical fatigue damage analysis using the American Welding Society model (high‐cycle fatigue analysis) to demonstrate the influence of the design stress level for fatigue‐critical members on the fatigue reliability and a first‐passage failure probability with periodic inspections. For the latter, a crack growth analysis is performed using the fracture mechanics method to estimate propagated crack size and corresponding residual strength under random service loading conditions. The effect of periodic inspections is very important if a fatigue crack already exists. It is also shown that there is always a limit in the number of inspections beyond which no significant improvement can be achieved.

Comparison of Gaussian closure technique and equivalent linearization method

Owing to their simplicity, the Gaussian closure technique and equivalent linearization method are frequently used in solving nonlinear random vibration problems. Whether Gaussian closure leads to more accurate result than the equivalent linearization method or vice versa is a controversial issue. Generally, they have the same analytical ingradients and will lead to identical or similar results depending on specific equivalent linearization techniques used. The present paper proves this assertion for relatively simple nonlinear systems. The method of proof, however, appears to be applicable to more complex problems. Examples are provided for a parametrically excited nonlinear system and a hysteric system.

Nonlinear ride analysis of heavy trucks

The problem of the dynamic interaction between the articulated vehicle and the road surface undulations is investigated using the equivalent linearization technique. The effects of the frictional forces generated in the laminated springs, bump-stops, wheel hop, road characteristics, loading condition and vehicle speed on the ride comfort and road safety are discussed and evaluated.

First passage probability and probability distribution of level crossings for a nonlinear system to nonwhite excitation.

The level crossing problem is discussed for a multi-dimentional nonlinear system subjected to an amplitude modulated nonwhite random excitation. The first passage probability and the probability distribution of the level crossings are approximately represented by a few low order factorial moments of crossings, which are calculated by using the equivalent linearization technique and assuming the Gaussian probability distribution of the response process. The dependence of occurrence of the level crossing upon the number of events is taken into consideration. Numerical results are compared with the corresponding digital simulation results for a single degree of freedom system with bilinear hysteresis excited by noise with an exponentially decaying harmonic correlation function.

Second order frequency domain analysis of moored vessels

A frequency domain analysis method for the dynamic response of a vessel moored in a random sea is presented, in which second order wave forces are included. The non-linearities occurring in the equations of motion of the vessel are accounted for by using the equivalent linearisation technique, leading to an iterative solution scheme for the mean values and covariances of the response. Application of the method to a moored drill ship indicates that the required computer time is relatively low. The method is intended for use in the preliminary stages of design where a parametric study may be required and the costs of a time domain analysis are prohibitive.

Vibrations of a system with non-linear hysteresis

A hysteretic model comprising a linear viscous part and a non-linear Davidenkov part is introduced in order to simulate rate- and amplitude-dependent internal losses during cyclic vibrations. The equivalent linearization technique is used to study the frequency response of a single-degree-of-freedom system. For an introduced model the problem of parameter identification is addressed. A vibrating cable is considered as an example for which theoretical frequency response curves and hysteretic loop shapes are compared with experimentally observed ones.

Modeling General Hysteresis Behavior and Random Vibration Application

A simple constructive technique for the development of rate-type hysteresis models for general nonlinear system is presented. The technique is used to develop hysteresis models to incorporate time history-dependent postyield restorting forces, and general pinching behavior in smoothly varying deteriorating models. Applications of these models to random vibration analysis modeling via simulation and equivalent linearization techniques under Gaussian noise excitation is presented.

Random vibration of hysteretic systems under bi‐directional ground motions

AbstractA random vibration method is developed for the response analysis of hysteretic structural systems under stochastic two‐dimensional earthquake excitations. The biaxial hysteretic restoring force is modelled by coupled non‐linear differential equations; the response statistics are obtained using the equivalent linearization technique. The validity of the proposed model is appraised using available biaxial loading tests of reinforced concrete columns. A parametric study was performed to examine the significance of the effects of biaxial interaction under earthquake excitations. A practical method to evaluate the extreme response statistics is also presented.

An equivalent linearization technique for two-phase flow instability analysis

An equivalent linearization technique has been developed to analyze two-phase flow instability oscillation with finite amplitude in the frequency-domain. A one dimensional slip flow model has been used to formulate equivalent linear equations, and implemented into a computer program. A simple boiling channel has been analyzed by the program. Results for very small amplitude oscillations agreed well with results by a detailed linear frequency-domain program. Analyses for larger amplitude oscillations gave more stable results. Limit cycle amplitudes have been calculated from the amplitudes at which channel characteristics change from unstable to stable as the amplitude is increased.

Finite element analysis of elastoplastic contact problems with friction

Fang, T. and Wang, Z. N., A Generalization of Caughey's Normal Mode Approach to Nonlinear Random Vibration Problems, to be published in AIAA Journal. Caughey, T. K., Equivalent Linearization Techniques, Journal of the Acoustical Society of America, Vol. 35, No. 11, 1963, pp. 1706-1711. that is taken to establish the non-interpenetration conditions between components D and W. The variables X$ X^^ represent the Cartesian coordinates of the material points in contact components D and W. The functional EM can be expressed as

Methods of non-linear stochastic dynamics for the assessment of structural fragility

Some of the external events which can significantly contribute to the overall risk of a nuclear power plant, give rise to a dynamic excitation of the structural components which form the plant. The computation of the risk associated with these external events requires an investigation of the behaviour of the structural components beyond the elastic limit. The stochastic nature of the excitation, then, leads one to deal with a non-linear stochastic dynamic problem. No general method of solution exists for such a problem when large structural systems are considered, although classical methods of propagating uncertainty have been successfully employed. This paper investigates the possibility of formulating an approach founded on a suitable equivalent linearization technique. In particular the authors make operative a new method of fragility analysis to be applied directly to the linearized system. The numerical example considers a framed structural component: its aim is to show the degree of accuracy that can be reached by the approach formulated in the paper.

A finite element–equivalent energy linearization technique for the analysis of nonlinear plate vibration

AbstractA simplified technique for the dynamic analysis of geometrically nonlinear plate structures is developed. The essence of this technique is the construction of a linear substitute of the nonlinear problem. The linear substitute problem is derived from an equivalence criterion which involves balancing the energies of the linear substitute model and the nonlinear model over one period of oscillation. The linearized equations are discretized by a finite element method, and solutions at different amplitudes are obtained numerically by an incremental‐iterative scheme. To verify the equivalent energy linearization approach, example problems consisting of the free and forced vibration of nonlinear circular plates with various boundary conditions are studied. All results are compared to theoretical and numerical solutions in the literature. In addition, the forced vibration results are compared to available experimental results. These comparisons tend to validate the assumptions made in the equivalent energy linearization procedure. The proposed method is found to be computationally more efficient than other available procedures.

Equivalent linearizations for practical hysteretic systems

Experimental testing of a friction damped base isolation system has indicated a need for a new model of friction damping and for an appropriate equivalent linearization technique. The model for the damping adopted is a combination of viscous damping, constant Coulomb friction and linear Coulomb friction. This model is incorporated into the equation of motion for a single-degree-of-freedom system and the exact solutions are given for free vibrations and for steady-state vibrations excited by a harmonic force. The exact solution is taken as a basis for an equivalent linearization technique that can be used in conjunction with conventional design spectra for a practical design of such a system.

A comparison of approximate techniques for non‐linear seismic soil response

AbstractSome equivalent linearization techniques, applicable to problems dealing with non‐linear seismic amplification of layered soils, are systematically treated and critically compared in this paper. The discussion concerns, in particular, the theoretical consistency, the operative modalities, the computational complexity and the numerical reliability of these methods.A comparison is made between the results furnished by some techniques among those examined and the results obtained through a step‐by‐step integration of the non‐linear equations of motion, assuming as a test‐problem the one‐dimensional amplification of a horizontally stratified soil deposit characterized by the Ramberg‐Osgood constitutive model.

履歴系の統計的等価線形化法

A stochastic equivalent linearization technique of bi-linear hysteretic characteristics is developed. Using this technique, an ordinary differential equation for the estimation of the second order moments of a sigle-degree-of freedom bi-linear hysteretic system subjected to non-stationary random excitation is derived. A comparison of two techniques-an equivalent linearization and a Fokker-Planck equation techniques-are made concerning the differences of the two sets of the ordinary differential equation and the transient r. m. s. displacement and velocity responses given by these two techniques. Extention of this approach to a multi-degree-of-freedom bi-linear hysteretic system and a single-degree-of-freedom poly-linear hysteretic system is discussed.