Abstract
A numerical order verification technique is applied to demonstrate that the asymptotic expansions of solutions of the Duffing equation obtained respectively by the Lindstedt-Poincaré(LP) method and the modified Lindstedt-Poincaré(MLP) method are uniformly valid for small parameter values. A numerical comparison of error shows that the MLP method is valid whereas the LP method is invalid for large parameter values.
Highlights
The Duffing equation u& & + ω 2 0 u +ε u3 =ε p cos Ωt has been used to model a number of mechanical and electrical systems [1]
The order of the asymptotic expansion solutions of free vibration of the Duffing equation has been verified in Ref.[6], but we note that the reversion method is adopted there and the consequent expansions contain the secular term ε t sin t, which are effective only for small values of ε t
We show a numerical comparison of the MLP method with the LP method
Summary
Ε p cos Ωt has been used to model a number of mechanical and electrical systems [1]. The differential equation that describes this oscillator has a cubic nonlinearity, and it has been named after the studies of G. A modified Lindstedt-Poincaré(MLP) method [3, 4] was proposed to obtain asymptotic expansion solutions of the Duffing equation, which works for small parameter values and for large parameter values of ε. A numerical order verification technique, first proposed by Bosley [5], will be applied to demonstrate that the asymptotic expansions of solutions of the Duffing equation are uniformly valid up to the third order for small values of parameter ε. The order of the asymptotic expansion solutions of free vibration of the Duffing equation has been verified in Ref.[6], but we note that the reversion method is adopted there and the consequent expansions contain the secular term ε t sin t , which are effective only for small values of ε t. A numerical comparison of the error of the LP method with that of the MLP method shows that the MLP method works for large values of ε whereas the LP method is invalid
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