An equivalent linearization technique to obtain the response of non-linear multi-degree-of-freedom dynamic systems to stationary gaussian excitations is developed. The non-linearities are assumed to be single-valued functions of accelerations, velocities and displacements. Using a property of gaussian vector processes, the closed forms of the coefficients of the equivalent linear system are obtained by the direct application of partial differentiation and expectation operators to the non-linear terms. It is shown that when the non-linearities possess potentials, the linear system has symmetric coefficient matrices. A geometrical interpretation of the linear coefficients, in connection with the original non-linearities, is presented. The accuracy is investigated by means of examples.

Stochastic linearization of multi-degree-of-freedom non-linear systems

http://link.springer.com/content/pdf/10.1007/978-3-642-83535-3_29.pdf

Stochastic Linearization in Structural Dynamics

An investigation is made of the possible application of tuned liquid column dampers and tuned liquid column/mass dampers in reducing the along-wind response of wind-sensitive structures. The structure is modeled as a lumped mass multi-degree-of-freedom system taking into account both bending and shear. The wind turbulence is modeled as a stochastic process that is stationary in time and nonhomogeneous in space. A random vibration analysis utilizing transfer matrix formulation is carried out to obtain response statistics. The nonlinear damping term in the fundamental equation of the tuned liquid damper is treated by an equivalent linearization technique. Numerical examples show that tuned liquid dampers, which have significant practical advantages, are as effective as the traditional tuned mass dampers if the parameters of the liquid dampers are properly selected. However, excessive liquid motion in a tuned liquid column/mass damper may reduce the effectiveness of this damper. It is also shown that the wind-induced force- and acceleration-type responses of the structure with a damper, which is usually tuned to the fundamental frequency of the structure, should involve more than one vibration mode as higher-mode responses may become as large or even larger than the controlled-mode response.

Control of Along-Wind Response of Structures by Mass and Liquid Dampers

Effectiveness of tuned liquid column dampers for vibration control of towers

A method is presented whereby a non-linear second order dynamical system is replaced by a linear system in such a way that an average of the difference between the two systems is minimized. Provided the averaging operator possesses certain properties, it is shown that the replacement is unique and can be accomplished in a straightforward manner. The parameters of the replacement linear system are expressed in terms of averages of functions of the linearized solution.

A generalization of the concept of equivalent linearization

Application of Statistical Linearization Techniques to Nonlinear Multidegree-of-Freedom Systems

This paper presents an approach which provides a particularly simple and direct way of determining the instantaneous correlation matrices for the stationary random response of multidegree-of-freedom linear systems subjected to excitations of nearly arbitrary spectral density. In the special case of white excitation, the instantaneous correlation matrices are determined directly from a set of linear algebraic equations. When the excitation is nonwhite, some integrals must be evaluated before solving a system of linear algebraic equations. However, the form of these integrals is considerably simpler than that encountered in other common approaches.

I-Min Yang

Papers

Application of Statistical Linearization Techniques to Nonlinear Multidegree-of-Freedom Systems

Calculation of Correlation Matrices for Linear Systems Subjected to Nonwhite Excitation