Sample Exam 2 w Solutions (2011) – Differential Equations

Problem 5. (10 pts) A piece with mass m inside a mechanical device is moving along anaxis and is subject to a force f(t), where f(t) = 23|t| (in Newtons) for −2 ≤ t ≤ 2 and fis periodic with period 4 s. The piece is also constrained by what is effectively a spring withspring constant 10 N/m. Friction is negligeable.Find all values of the mass m such that the piece could enter into resonance.The natural frequency of the piece is ω 0 =√10mrad/s. The piece can enter into resonance ifand only if ω 0 is also a frequency of the external force f(t). Let us find those frequencies bycomputing the Fourier series of f(t) (or rather |t| since a constant factor doesn’t change thefrequencies). Note that |t| is even, so there are no sine terms:|t| = A ∞02 + ∑A n = 2 2= 2nπ∫ 20n=1A n cos nπt2tcos nπt2 dt([tsin nπt2 ]2 0 −= ( 2nπ )2 [cos nπt2 ]2 0= ( 2nπ )2 ((−1) n −1){0, n even=−2( 2nπ )2 , n odd.∫ 20sin nπt2 dt )The frequencies in f(t) are therefore ω n = nπ 2for the following values of m:for all odd integers n > 0. Resonance can happenω 0 = ω nwhere n is an odd integer.√10m = nπ 210m = (nπ 2 )2 = n2 π 24m = 40n 2 π kg 210