THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

C.5 Equations with Complex Roots 645Thenx(t) = M(e −αt+iβt+iφ + e −iαt−iβt−iφ ) + addtional terms= Me −αt (e i(βt+iφ) + e −i(βt+φ) ) + additonal terms= 2Me −αt (ei(βt+iφ) + e −i(βt+φ) )+ additonal terms2= 2Me −αt cos(βt + φ) + additional termsHere M and φ can be determined graphically using Figure C.3. The use of thegraphical residue technique for equations with complex roots is extremely powerfulsince the inverse transform can be determined by inspection.Example Problem 4GivenX(s) =find the inverse Laplace transform.Solutions + 2s(s 2 + 4s + 5)Figure C.4 shows the function. The residue at the pole at the origin is found byusing Figure C.4(b), and the complex poles are derived by using Figure B.4(c).The ensuing function isx(t) = √ 2 √ + 2 × 15 5 2 √ 5 e−2t cos(t + φ)whereinφ = (90 ◦ ) − (90 ◦ + θ)θ = ( π − tan −1 )12Example Problem 5From vibration theory X(s) [cf. Equation (20.12)] is of the forms + 2ξω nX(s) = x 0s 2 + 2ξω n s + ωn2 x 0 (s + 2ξω n )=√ √(s + ξω n + iω n 1 − ξ 2)(s + ξω n − iω n 1 − ξ 2)Determine the inverse transform.