Its derivation involves only matrix algebra and some properties of the instantaneous correlation matrices of a. The theory of the random process was first successfully applied to dynamical systems by Einstein 1. He used it to investigate the Brownian motion of a free particle and found that the probability density of the reaction process is governed by a diffusion equation.
Since the spectral density of the steady-state response of a linear system can be easily determined by Fourier transform, Wiener-Khintchine. Unfortunately, only one of the three approaches mentioned above, the Fokker-Planck approach, can be extended to nonlinear problems. Additionally, as a byproduct of the nonlinear analysis, a matrix algebra approach is developed that can be used to find various instantaneous correlation matrices for the stationary random response of a linear system with multiple degrees of freedom.
Two examples are worked out in detail to illustrate the application of the matrix algebra approach. The new approach for determining the instantaneous correlation matrices of the steady-state response of a multi-degree-of-freedom nonlinear system is presented in Chapter III, along with a brief summary.
STATIONARY RANDOM RESPONSE OF MULTI-DEGREE-OF-FREEDOM
The first two approaches can be used to calculate the mean vector and matrix of the correlation function of the response process. In the third approach, the Fokker-Planck equation, which governs the transition probability density of the response process, can only be used if the excitation is a shot or white noise. 1 7) we know that the spectral density function matrix of the response vector x is given by.
Therefore, the matrix of the frequency response function at w = 0 is exactly the inverse of the stiffness matrix. In this case, the instantaneous correlation matrices of the stationary response are governed by the following equations (see Appendix A for details). 5 5) Therefore, the current covariance matrix x is equal to the current correlation matrix if the mean excitation vector.
Analogous to the second order system (2. 1 ), we define the impulse response function matrix G(t) of (2. 6 7) as the solution of the following system. In the second part of this chapter we discussed the impulse function approximation, the spectral density approximation and the Fokker-Planck approximation. In this case, both the Fokker-Planck approach and the matrix algebra approach lead to the same equations for determining the.
If the excitation is a white Gaussian noise, then the transient probability density of the response process is governed by the Fokker-Planck equation. In this approach, the nonlinear terms of the system are required to be small compared to the linear terms and the excitation level must also be quite low. It should be noted that the terms in square brackets are the kinetic energy and the potential energy of the system, respectively.
M~ + C(o)5( +K<0>x +µg(X, x) = f(t), µ=a small parameter (3 .11) The matrices C(O) and K(O) are respectively the damping matrix and the stiffness matrix of the system due to the linear part of the damping forces and spring forces, and µg(x, x) represents the nonlinear forces of the system. Equations (3. l 7) and (3 19) can be used to find various root mean square values of the response process Let K(e) and C(e) denote the stiffness matrix and the damping matrix of the linear system defined by (3. 38).
Two types of nonlinear springs will be considered after the discussion of the approximate solution. It is clear that this approach is only valid if both the nonlinearity of the system and the excitation are sufficiently small.
EXAMPLES
For example, buildings subjected to severe excitations often behave like lightly damped damping systems, i.e. the stiffness of the system decreases as the displacement increases. In addition, the effective damping in the first few modes is often only a few percent of the critical damping. The dampers are non-linear and their damping force is proportional to the square of the relative velocity.
4 is a truncated Gaussian white noise with cutoff frequency w c • Here the mean square displacement is plotted against w • When w is not close to prime. It is therefore clear that for a Gaussian white noise excitation the largest contribution to the mean square displacement comes from the first mode of the linear equivalent system and the contributions from the higher modes are almost negligible. This indicates that the equivalent linear damping in the higher modes is much larger than that in the first mode.
To gain more insight into the damping behavior of the nonlinear system, it is informative to consider a linear system that has the same M, K, f(t ), and instantaneous correlation matrices as the nonlinear system (4.3). 5 we can compare the nonlinear system with a linear system that differs from the nonlinear system only in the dampers used. Thus, if the spectral density varies slowly in these neighborhoods, then the excitation can be approximated by a white noise whose spectral density is just equal to that of the original excitation at w = w.
In this way, the analysis of an equivalent linear system can be greatly simplified because the steady-state response of the linear system1 is subjected to a. White noise excitation can only be found by solving a system of linear algebraic equations. The linear dampers are arranged so that the equivalent linear system for the nonlinear system has normal modes and the damping in each mode is 5% of the critical damping. 7 shows that as the excitation level increases, the rms displacement of the first spring increases much faster than the second and third springs.
These figures show the mean square displacements of the first and third springs as a function of w, respectively. However, for springs other than the first, the mean square displacement in the first mode may be similar to that in the other modes. 10 it is seen that for the third spring the contribution of the second mode can be of the same order of magnitude as that of the first mode.
SUMMARY AND CONCLUSIONS
It requires that the linearized system have normal modes and that the correlation function matrix of the excitation is diagonalized by the same matrix that decouples the linearized system. The first condition may not be too serious, but the second condition regarding the excitation makes the application of this approach quite limited. In the present study, a generalized equivalent linearization approach is used to determine the instantaneous correlation matrices of the stationary random response of a non-multi-degree-of-freedom model.
After applying the matrix algebra approach to the equivalent linear system and the special iteration scheme described in Chapter III, one is led to the iterative solution of a system of linear algebraic equations. It is implicitly assumed that to obtain a good approximation solution, the nonlinearities of the system must be small. Two examples that can be solved precisely with the Fokker-Planck approximation have also been worked out with the generalized equivalent linearization approach.
A comparison of the results shows that for a cubic hardening system the error in the mean square displacement is always within 7.5% of the exact solution and that the largest error for an arctangent softening system is 11. Two more examples which can be solved by the new approach given in Chapter I IV.
In the first case, linear springs and shock absorbers were used, the damping force of which is proportional to the square of the speed. The results show that the dampers used in this case are particularly suitable for systems subject to strong excitation. The linear dampers were arranged so that the linear system is equivalent to the nonlinear one.
Since the system (A. 1) is assumed to possess a stationary response, therefore all eigenvalues will have a negative real part. From the definition of the characteristic function and properties of a Gaussian distribution, it is known that u is Gaussian distributed with zero mean and its variances and covariances are given by. Let x(t) be a stationary random vector process which ·is assumed to be differentiable in the mean square with the required order.
EXACT
2.,, · Spectral density in
25 Cutoff
5 frequency in
Cutoff
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