fa_09lecture_set5a

# fa_09lecture_set5a - Periodic Solutions(Jordan and Smith...

This preview shows pages 1–5. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Periodic Solutions (Jordan and Smith, Chap. 5) Consider equation For small motions, we can write as ( see how & introduced ): Note that for & =0, 2 sin( ) cos . (5.1) x x τ + Ω = Γ && 2 3 cos . (5.14) x x x ε τ + Ω- = Γ && & Assuming a power series solution equation (5.14) gives Collecting terms of powers of & gives O( ): O( ): O( ): ε 1 ε 2 ε For 2 &-periodic solutions, which gives Solutions of (5.17a) are (assuming ¡ ¢ integer): & For solutions (5.21) to be 2 &-periodic, and Then, (5.17b) becomes: The only 2 &-periodic solution is 0, a b = = & Thus, a two-term approximation is: The above is called the non-resonant case . Example 5.1 : Consider the system We have to introduce a small parameter & to use perturbation methods. So, we embed the system into the equation & Now, assume soln: Since the forcing is 2 ¡-periodic, let us seek solutions with this periodicity. Then, and the sequence of problems are: : :...
View Full Document

{[ snackBarMessage ]}

### Page1 / 15

fa_09lecture_set5a - Periodic Solutions(Jordan and Smith...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online