Question

3.2.2 Resonance At this point, we can rewrite tan 6 as tan eda (40) and Fo/a (41) where (42) a. is the resonant frequency. This is because the particular solution, or resp greatest magnitude when the driving frequency equals this value. In other words, the system res- onates with maximum response at this frequency onse function, has the Exercise 30 Plot the expression for |B|, as provided by Eqn. 41, vs. wa. Show that this quantity has as its peak inax (43) b2 bu 4ac where 62 2ac (44)

Answer 1

dB/dw_d = d/dw_d (-(F_0/a)/squareroot (w^2_d - w^2_R)^2 + (bw_d/a)^2) dB/dw_d = -(F_0/a) (-1/2) (squareroot (w^2_d - w^2_R)^2 + (bw_R/a)^2)^-3/2 (2(w^2_d - w^2_R) (2w_d) + 2 b^2/a^2 w_d dB/dw_d Vertical-bar _1_d = w_max = 0 \r\n +(F_0/a) [2(w^2_d_max - w^2_R) w_d + b^2 w_d/a^2]/(squareroot (w^2_d_max - w^2_R)^2 + (bw_d/a)^2)^2 = 0 2(w^2_d_max - w^2_R) + b^2/a^2 = 0 (w^2_d_max - w^2_R) + b^2/2a^2 = 0 w^2_d_max = w^2_R + b^2/2a^2 = c/a + b^2/2a^2 w^2_d_max = c/a [1 + b^2/2ac] w_d_max = squareroot c/a squareroot 1 + b^2/2ac w_d_max = w_R squareroot 1 + b^2/2ac asymptoticallyequalto w_R Vertical-bar B Vertical-bar _max = (F_0/a)/squareroot (w^2_R - w^2_R)^2 + (b w_R/a)^2 = F_0/a/bw_R/a asymptoticallyequalto F_0/bw_R