Equivalent Linearization for Nonlinear Random Vibration Elishakoff, I. And Cai, G.Q., “Approximate...

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Equivalent Linearization for Nonlinear Random Vibration Elishakoff, I. And Cai, G.Q., “Approximate solution for nonlinear random vibration problems by partial stochasti c linearization”, Probabilistic Engineering Mechanics, V ol. 8, pp. 233-237,1993. Zhao, L. and Chen, Q., ”An equivalent non-linearization method for analyzing response of nonlinear systems to ra ndom excitations”, applied Mathematics and Mechanics, Vo l. 18, pp. 551-561, June 1997. Polidori, D.C. and Beck, J.L.,“Approximate solutions for non-linear random vibration problems”, Probabilistic Eng ineering Mechanics, Vol. 11, pp. 179-185, July 1996.
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  • Equivalent Linearization forNonlinear 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. 233-237,1993.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. 551-561, June 1997.Polidori, D.C. and Beck, J.L.,Approximate solutions for non-linear random vibration problems, Probabilistic Engineering Mechanics, Vol. 11, pp. 179-185, July 1996.

  • ContentsEquivalent Linearization Method for SDOF SystemPartial Stochastic Linearization MethodEquivalent Non-linearization MethodApproximate Solutions for Non-linear Random VibrationEquivalent Linearization Method for MDOF System

  • 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 systemAn approximate solution can be obtained from the followinglinearized equationDefineThen

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

  • 1.3 Low Non-linear System Consider a non-linear system with the following formThe solution isThe results from the equivalent linearization method andthose from the perturbation method agree to the first orderin e.

  • 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 be and Ke. 2. Calculate the response of the equivalent linear system. 3. From the response, calculate the new be and Ke. In certain simple cases, its possible to obtain an explicit expression for sX . In general, the accuracy of the first and thesecond statistical moment based on the method can satisfy thedemands 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 equationThe equivalent equation with linear damping forceThe criterion for selecting is that average energy dissipation remains the same

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

  • Illustrative ExampleConsider a system governed byThe stochastic linearization method yieldsandThe partial linearization method yields

  • 3. Equivalent Non-linearization MethodReplacing 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 FPK

  • Example 3.2 FPK

  • 7/3231/27/3137/3!!

  • Example 3.3

  • Example 3.4

  • 4. Approximate Solutions for Non-linear Random VibrationEquivalent linearization methodDefineThenFor 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 equationsThe probabilistic linearization technique finds the linear system whose PDF, , best approximates eq(2), i.e.(2(3)where well use the standard or a weighted L2 norm.()

  • Given any two functions f,g : RnR, the standard L2 inner product of the functions is defined bySimilarly, given any weighting function , an inner product can be defined byThe standard L2 norm of a function is and a weighted L2 norm can be defined by

  • Example: Linearly Damped Duffing OscillatorConsidered the non-linear system :Equivalent linearization

  • Using the standard L2 norm :Using the L2 norm weighted with (1+y12) :Probabilistic linearization

  • E[x2] for smallE[x2] for larger

  • 5. Equivalent Linearinzation Method for MDOF SystemsConsider 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

  • 3n23n2

  • 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

    The technique is based on the concept of replacing the given nonlinear system by a related equivalent linear system in such a way that the statistical difference between the two systems is minimized. And an approximate solution of the given nonlinear system may be obtained by means of solving the equivalent linear system.It should be pointed out that the smallness of the nonlinear term in the original equation was not a presupposition in the development of the equivalent linearization method. Wolaver has shown that in the case of a system where only the stiffness law is nonlinear and where the excitation is a Gaussian white noise, the equivalent linearization method leads to the exact mean-value for the stationary displacement regardless of the magnitude of the nonlinear term.However, it must be noted that the results determined by the method are usually smaller than those by FPK equation method of Monte Carlo simulation, and maximum error is on the low side about 20 percent. Hence, the error is obvious too big for structural systems with the higher accuracy. If the method were applied to structural reliability analysis, the results should have led to dangerous error.Only the nonlinear damping force in the original system is replaced by a linear viscous damping, while the nonlinear restoring force remains unchanged. The replacement is based on the criterion of equal mean work.Where f represents the non-linear damping force which is replaced by an equivalent linear damping force and h represents the non-linear restoring force which remain unchanged.

    For the system with non-linear damping but with linear restoring force, the present method coincides with the usual linearization method. For the system with both nonlinear damping and nonlinear restoring force, it represents a natural generalization of the stochastic linearization.

    The left and right hand side of the equation represents the average work pre unit time preformed by the original nonlinear damping force and the equivalent linear damping force, respectively.Although the identical criterion is used in both methods, the ensemble averaging is performed with different probability density function. In the stochastic linearization, the probability density is assumed to be Gaussian, while in the present partial linearization method, it is generally not Gaussian. Its worthy to note that the formula is not the explicit solution to the equivalent damping coefficient.Making use of the fact that the exact stationary probabilistic solution of the equivalent partial linearization system is equal to pK.This is a nonlinear algebraic equation for the damping coefficient.

    Fig. 1 Mean square displacement with respect to stiffness nonlinearity; K=1, b=0.1, a=0.5, r=1Fig. 1 Mean square displacement with respect to damping nonlinearity; K=1, b=0.1, d=1, r=1

    As expected, the partial linearization method yields more accurate results than the equivalent linearization since it remains one of the characteristics of the original nonlinear system, namely the nonlinear restoring force. Except that, the partial linearization method requires less computation.System with nonlinear stiffness (cubic hardening spring system)Duffing systemSystem with nonlinear dampingSystem with nonlinear damping and stiffnessFor each of these described methods, approximations are made to obtain a simpler system that provides the best fit in some sense to the non-linear system of interest. Alternative approach for finding an equivalent linear system are presented here. The proposed method differs from existing ones in the criteria used to measure the best fit.Note that we have avoided solving the FPK equation; computing the norm amounts to differentiating a Gaussian probability density function with the linear operator L(y) and then integrating the results. Future work includes testing the methods on multiple degree of freedom systems and investigating what weighting function should be used in the norms.