Undamped Forced Systems We now look at the pure resonant case for a second-order LTI DE. We will use the language of spring-mass systems in order to interpret the re sults in physical terms, but in fact the mathematics is the same for any second-order LTI DE for which the coefﬁcient of the ﬁrst derivative is equal to zero. The problem is thus to ﬁnd a particular solution the DE x �� + ω02 x = F0 cos ωt. The steps, as in the example in the last note, are Complex replacement: z�� + ω02 z = F0 eiωt , x = Re(z). Characteristic polynonial: p(r ) = r2 + ω02 ⇒ p(iω ) = ω02 − ω 2 . ⎧ F0 eiωt F0 eiωt ⎪ ⎪ = ⎪ ⎨ p(iω ) ω02 − ω 2 Exponential Response formula ⇒ z p = ⎪ F0 teiωt ⎪ F0 teiωt ⎪ ⎩ � = p (iω ) 2iω ⎧ F cos ωt 0 ⎪ ⎪ if ω �= ω0 ⎨ ω2 − ω2 0 ⇒ xp = ⎪ ⎪ ⎩ F0 t sin ω0 t if ω = ω0 (resonant case). 2ω0 if w �= ω0 if ω = ω0 . Resonance and amplitude response of the undamped harmonic oscillator F0 In x p the amplitude = A = A(ω ) = | 2 | is a function of ω. ω0 − ω 2 The right plot below shows A as a function of ω. Note, it is similar to the damped amplitude response except the peak is inﬁnitely high. As w gets closer to ω0 the amplitude increases. F0 t sin ω0 t When ω = ω0 we have x p = . This is called pure resonance 2ω0 (like a swing). The frequency ω0 is called the resonant or natural frequency of the system. In the left plot below notice that the response is oscillatory but not periodic. The amplitude keeps growing in time (caused by the factor of t in x p ). Note carefully the different units and different meanings in the plots below. Undamped Forced Systems OCW 18.03SC The left plot is output vs. time (for a ﬁxed input frequency) and the right plot is output amplitude vs. input frequency. x and A are in physical units dependent on the system; t is in time; ω is in radians. x A t ω0 Resonant response (ω = ω0 ) ω Undamped amplitude response 2