# New PDF release: A Course of Mathematics for Engineers and Scientists. Volume

By Brian H. Chirgwin and Charles Plumpton (Auth.)

ISBN-10: 0080131328

ISBN-13: 9780080131320

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Additional resources for A Course of Mathematics for Engineers and Scientists. Volume 5

Example text

A general method of finding inverse S 2:1 THE LAPLACE 45 TRANSFORMATION TABLE T L {/(*)} /(*) 1 1. 1 2. tn, n an integer > 0 pTi+i 3. eat, a < p 1 p —a 4. tn eat, a < p, 7i an integer ^ 0 5. sin ω t 6. coswt 7. t sin oj t 8. t cos ω t 9. sinhai a ρ2 — α2 10. _ ρ2 — α2 11. sin(o)2 -f- a) 12. cos(a>2 -(- ß) V n\ n\ (p ~ a)71*1 p2 + ω2 _ ^ j92 + ω 2 2ω£> 1 2ω2 2 2 = 2 # 2 + ω 2 ~ (ί) +~^ ) V sin a î>2 (ρ2~ω*)* + ω cos a + ω2 V cosß ^2 ρά — ω ' + cosinß ω2 transforms, using t h e Fourier-Mellin inversion theorem of contour integration will be given in Chapter I V .

This result is a special case of a very general principle (see Vol. I l l , § 9:6). If a vibration of an oscillating system is resolved into the sum of component vibrations in the normal modes, the kinetic energy of the vibration is the sum of the kinetic energies of the component vibrations in the normal modes. , there are no terms involving products of different normal coordinates. Miscellaneous Exercises I 1. Prove t h a t the Fourier expansion of the function x2 in the interval — π <; x ^ π is oo 4 â n2 + Σ (— l) w —£ oosnx.

Mi Therefore 2» 2 + 5 » + 1 χ =>(2> — F +τ l ) F( Pτ + 2 ) = Therefore (iii) Solve the equation where # 0 — 0, xx = 0. 1 2 2 p 4- -p + 1 1 2(^+2) x = J + 2e-' — ie~2i. (p2 ap + ω2) (p2 + n2) ' Two cases arise: (1) n φ ω. Then V n 2 — ω 2 \ p2 + ω 2 x V p2 -\- n2 j ' The inverse transformation gives X = α (cosoüi — cos ni) n2- ω2 ' (2)TO= ω. Then (ί>2 + ω2)2 * The inverse transformation (Table I, no. 7) gives at x = ——einwt. 2ω (iv) Solve the equation d3x di 3 d2x dt2 dx + x = St e- 1 , dt where # 0 = 0, α^ = 1, x2 = 0.