Z] ^^k COS (ojt + e^) 



Ogilvie 



■^'(^■"jk + Z^jk^ + Cjk + 





(t) sin tiiT dr 



+ ) a^ sin (oJt + £■ ) 



-ojb 



jk 



•J n 



(t) cos 6JT dr 



6 /-CD 



E Vt) Pijk + /^jk - i:^ Kj^(T) sin c^T dr 



E -^k^t) |b.^ + Kj^(r) cos cr dr + ^ a^(t) Cj, 



k = l •• -'0 -^ k = 1 



(14) 



We can identify this expression directly with the left-hand side of (4), and fur- 

 thermore it is clear that the right-hand side of (13') will be identical with the 

 right-hand side of (4).* Thus Eq. (13') reduces to (4) in the special case of 

 sinusoidal oscillations. 



We note specifically that the term 



r 



CO I ^i^.i'^) sin cl^t dr 



in (14) could just as well have been combined with the term c ^ as with /i.^. 

 However, as mentioned previously, this ambiguity is usually resolved by includ- 

 ing in the displacement force (i.e., in the sum over aj^(t)) only the zero-fre- 

 quency contributions. The only part of the coefficient of cos (cot + e^) which is 

 non-zero when w^o is c ■, , and so we let it stand alone. 



jk' 



Thus we have seen that the time- and frequency -do ma in descriptions are 

 equivalent if all functions depend sinusoidally on time. The same is true for 

 non- sinusoidal disturbances. We show this simply by taking Fourier transforms 

 of Eq. (13'). Suppose first that the disturbance is a transient such that all mo- 

 tions die out after a reasonable time and all displacements approach zero (at 

 least asymptotically). Then we can take Fourier transforms of (13'), obtaining: 



E -^'^"^jk + ^jk) + i^b.^ + (c.^ + mg/3^S.^) + icu3{Kj^} 9 {a J 

 k = i L -I 



. 3{F.+G.}, 



where 



*The definitions of Cj^ are slightly different in (4) and (13'), the effect of the 

 "pendulum" terms, mg /3. a.(t), being included in c . .a. ( t ) in (4). 



36 



