Sample Exam 2 w Solutions (2011) – Differential Equations

Problem 2a. (9 pts) A mass-spring-dashpot system with mass 1 kg, spring constant 4.5N/m, and damping coefficient 1 N/(m/s) is subject to an external force F(t) = 10cosωt. Findthe steady periodic solution x sp (t).The equation is x ′′ +x ′ +4.5x = 10cosωt. The steady periodic solution is the solution of thenon-homogeneousequationwhichdoesnotcontainexponentials. Wefinditusingundeterminedcoefficients:x sp = Acosωt+Bsinωtx ′ sp = −Aωsinωt+Bωcosωtx ′′sp = −Aω 2 cosωt−Bω 2 sinωtx ′′sp +x ′ sp +4.5x sp = (−Aω 2 +Bω +4.5A)cosωt+(−Bω 2 −Aω +4.5B)sinωt= 10cosωt{(4.5−ω 2 )A+ωB = 10−ωA+(4.5−ω 2 )B = 0[ ][ ] [ ]4.5−ω2ω A 10−ω 4.5−ω 2 =B 0[ [ ] A 4.5−ω2 −1 [ ]ω 10=B]−ω 4.5−ω 2 0[ ][ ]1 4.5−ω2−ω 10=(4.5−ω 2 ) 2 +ω 2 ω 4.5−ω 2 0[ ]10 4.5−ω2=.(4.5−ω 2 ) 2 +ω 2 ωThe steady periodic solution is:x sp (t) =10(4.5−ω 2 ) 2 +ω 2 ((4.5−ω 2 )cosωt+ωsinωt ) .4