I really need help solving this problem with necessary steps
explained. Below I will attach Eq 4.12, 4.14, and 4.15 for
reference
I Really Need Help Solving This Problem With Necessary Steps Explained Below I Will Attach Eq 4 12 4 14 And 4 15 For 1 (49.38 KiB) Viewed 28 times
These are the equations for reference
I Really Need Help Solving This Problem With Necessary Steps Explained Below I Will Attach Eq 4 12 4 14 And 4 15 For 2 (4.44 KiB) Viewed 28 times
Consider the Duffing oscillator with an equation mä = -r.i – ko(1+B.c?).. + Fo coswt This problem is not solvable because of the non-linearity introduced by the parameter B. You can set up a perturbation theory for this problem as follows; write x(t) = x1(t) + B.x2(t) +Bx3(t) +..., a Taylor expansion in powers of B. Substitute this expression into the Duffing equation and by collecting the terms of a given order in ß, write the linear equations satisfied by 21,12,13. As an example the lowest order equation for n=1 reads 1 - mči +rii + kolj - Fo coswt = 0. + -
The equations for 22 will involve 21 but not 13, and so forth. { This problem is an exercise in setting up perturbation theory. We dis- cussed this method in class in the context of solution of Eq (4.12) for the non-linear driven oscillator. You can think of this problem as finding the equation corresponding to Eq. (4.14) in T-M, without worrying about the solution in Eq. (4.15). One difference is that we want the equation for 13 as well as 22. }
ä = -ax + εx3 + G cos wt (4.12)
ä, fan - es - c)cos we + c (C 3 EA 4 1 &A cos 30t 4 (4.14)
1 3 X2 - (ad - *** - c)cos at - aA EA3 - G ot EAS 3602 cos 30t (4.15) W2 4
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