Equivalent Linearization for Nonlinear Random Vibration Elishakoff, I. And Cai, G.Q., “Approximate solution for nonlinear random vibration problems by partial stochastic linearization”, Probabilistic Engineering Mechanics, Vol. 8, pp ,1993. Zhao, L. and Chen, Q., ”An equivalent non-linearization method for analyzing response of nonlinear systems to random excitations”, applied Mathematics and Mechanics, Vol. 18, pp , June Polidori, D.C. and Beck, J.L.,“Approximate solutions for non-linear random vibration problems”, Probabilistic Engineering Mechanics, Vol. 11, pp , July 1996.

Contents 1.Equivalent Linearization Method for SDOF System 2.Partial Stochastic Linearization Method 3.Equivalent Non-linearization Method 4.Approximate Solutions for Non-linear Random Vibration 5.Equivalent Linearization Method for MDOF System

1.Equivalent Linearization Method for SDOF System 1.1 Origin * Originated by Krylov and Bogoliubov (1937) for the treatment of nonlinear systems under deterministic excitation * Bootom and Caughey (1963) first applied the method to random oscillation problems

1.2 Derivation For a non-linear SDOF system An approximate solution can be obtained from the following linearized equation Define Then

If F(t) is stationary, Gaussian, and has a zero mean, then The undetermined coefficients can be rewritten as Note: The formulas are not explicit expression for  e and Ke, since the expectations appearing on the r.h.s depend on  e and Ke.

1.3 Low Non-linear System Consider a non-linear system with the following form The solution is The results from the equivalent linearization method and those from the perturbation method agree to the first order in .

1.4 Highly Non-linear System When the non-linear term is not small, the equivalent linearization is often an iterative procedure. 1. Guess an initial value of  e and K e. 2. Calculate the response of the equivalent linear system. 3. From the response, calculate the new  e and K e. In certain simple cases, it’s possible to obtain an explicit expression for  X. In general, the accuracy of the first and the second statistical moment based on the method can satisfy the demands of engineering application.

2. Partial Stochastic Linearization Method Classical stochastic technique, where both nonlinear damping and nonlinear restoring force are replaced by their respective linear counterparts. Partial Linearization using the concept of average energy dissipation to deal with the part of nonlinear damping only.

Consider the following nonlinear equation The equivalent equation with linear damping force The criterion for selecting is that average energy dissipation remains the same

According to the Ito differential rule and take the ensemble average, we can get K is the spectral density of the white noise excitation F(t) Through some tedious derivation, can be solved analytically or numerically.

Illustrative Example Consider a system governed by The stochastic linearization method yields and The partial linearization method yields

3. Equivalent Non-linearization Method Replacing the non-linear restoring function by an g by an equivalent linear damping force and a non-linear restoring force. To minimize the difference between the two systems

We assume that the excitation F(t) is stationary and Gaussian white noise and has a zero expectation. The velocity and the displacement are independent of each other. The formulas are not explicit expression for and. Hence, an iterative solution procedure is generally required to select the desired and.

Example 3.1 當系統為 時， 可以與 FPK 方 法得到的解析解比較，知道此方法所得到的結果 比等價線性法誤差小

Example 3.2 同樣，在當系統為 時， 與 FPK 方法得到的解析解比較，結果也是比等價線性 法誤差小

如右圖所示，若以 7/3 次方來近似高次方項， 例如： 2 次或 3 次，則 必然可以得到比使用 線性更好的結果。 若此非線性系統與 1/2 次方相關，則顯然 7/3 次方並不會得到比線 性近似更好的結果。 ½ 次方 1 次方 3 次方 7/3 次方 這就是破綻 !!

Example 3.3 若以此方法模擬的非線 性阻尼系統，效果也不錯

Example 3.4 最後當系統為時，雖 然結果比等價線性法好，但就高度非線性的問題 而言，誤差仍然相當大

4. Approximate Solutions for Non-linear Random Vibration Equivalent linearization method Define Then For a non-linear SDOF system

Let & be the forward Kolmogorov operators corresponding to the non-linear & the linear system, respectively, and & be the solutions to the FPK equations The probabilistic linearization technique finds the linear system whose PDF,, best approximates eq(2), i.e. …… (2 …… (3) where we’ll use the standard or a weighted L 2 norm. …… ( ☆ )

Given any two functions f, g : R n  R, the standard L 2 inner product of the functions is defined by Similarly, given any weighting function, an inner product can be defined by The standard L 2 norm of a function is and a weighted L 2 norm can be defined by

Example: Linearly Damped Duffing Oscillator Considered the non-linear system : Equivalent linearization

Using the standard L 2 norm : Using the L 2 norm weighted with (1+y 1 2 ) : Probabilistic linearization

E[x 2 ] for smallεE[x 2 ] for largerε

5. Equivalent Linearinzation Method for MDOF Systems Consider a non-linear MDOF system where M,C,K denote constant nxn matrices  is a non-linear n-vector F is a non-linear n-vector of excitation The equivalent linear system is where Me,Ke,Ce are deterministic matrices.

Define the error Minimize the error min If we let then we can derive

上式共有 3n 2 個未知數， 也有 3n 2 個未知數

When the excitation is Gaussian This set of non-linear equations must be solved iteratively. For instance, the procedures can be ini- tialized by neglecting  and solving