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Unformatted text preview: Nonlinear Dynamic Systems Multiple Scales Analysis The method of multiple scales considers an expansion that represents the response of the system to be a function of two or more independent variable (i.e. multiple time scales). For instance, the new independent time scales are related by τ = t and τ n = n t for n = 1 , 2 . . . , (1) where t is the actual time, is a small non-dimensional parameter, and each τ n represents a different time scale in the response of the system. This causes a change in the derivatives with respect to time d dt = ∂ ∂τ dτ dt + ∂ ∂τ 1 dτ 1 dt + ∂ ∂τ 2 dτ 2 dt = D + D 1 + 2 D 2 , (2a) d 2 dt 2 = D 2 + 2 D D 1 + 2 2 D D 2 + 2 D 2 1 . (2b) The common approach is that one assumes a solution in the form of an expansion x ( τ, ) = x ( τ, τ 1 , τ 2 ) + x 1 ( τ, τ 1 , τ 2 ) + 2 x 2 ( τ, τ 1 , τ 2 ) , (3) where the number of terms in the expansion is equivalent to the number of independent time scales. 1 Quadratic restoring force This section examines a system described by Eq. (4) and compares the results of a first order expansion to the results of a secorder multiple scales expansion. ¨ x + 2 ζ ˙ x + ω 2 x + k 2 x 2 = 0 , (4) where ζ is the damping ratio, ω is the system linear natural frequency, and k 2 is a nonlinear coefficient of a quadratic nonlinearity. To apply the method of multiple scales, it is convenient to introduce a change of variable in the above equation 1 ¨ x + 2 μ ˙ x + ω 2 x + βx 2 = 0 , (5) where ζ = μ and k 2 = β . 1.1 First order expansion: O ( 1 ) The assumed solution to Eq. (22) is written as a first order expansion x ( τ, ) = x ( τ, τ 1 ) + x 1 ( τ, τ 1 ) , (6) where the independent time scales are defined as τ = t , τ 1 = τ . It follows that the derivatives with respect to time become the following expansion terms of the partial derivatives with respect to the corresponding time scale....
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This note was uploaded on 05/19/2011 for the course EML 4220 taught by Professor Chen during the Spring '08 term at University of Florida.
- Spring '08