446.328 Mechanical System Analysis 기계시스템해석 ‐ lecture 19, 20, 21‐ Dongjun Lee (이동준) Department of Mechanical & Aerospace Engineering Seoul National University Dongjun Lee Frequency Response *** need to first start with math. def. of H(jw); then its meaning for transfer function (input‐to‐output) stable steady‐state H(s) H(s) input transfer function output ‐ often, we want to know (or design) system’s response when the input signal contains certain frequency components, e.g., sin sin ⋯ excitation mode ‐ e.g., microphone/headphone, engine mount, earthquake‐proof structure,…) frequency response of Dongjun Lee 1 Frequency Response of TF transfer function (input‐to‐output) H(s) input transfer function output sin sin ∠ for stable , in steady‐state gain same frequency phase Im as input ex) 3 2 sin → sin tan 3 10 2 sin2 → sin ∠ 6 2 ex) 2 Re ∑ sin →∑ sin ∠ * if is stable and in steady‐state Dongjun Lee Derivation of Frequency Response ⋯ if is stable (i.e., all poles are in LHP), in steady‐state, 2 2 Im → ∠ → sin Re Dongjun Lee 2 Bode Plot transfer function (input‐to‐output) H(s) input transfer function output ‐ steady‐state frequency response when is stable ‐ gain plot: log vs 20 log [dB] (decibel) ‐ phase plot: log vs ∠ ‐ decade: 10 times of (e.g., from 5 [rad/s] to 50 [rad/s]) ‐ decibel: 20 log → allows us to “add” Bode plots 0 [dB] → 1 no amplification/attenuation 10 [dB] → 10. 3 → 3times amplification 10 [dB] → 10. 0.3 → attenuation 3 [dB] → 10/ 1/ 2 ‐ if cos 0 , → 0 cos∠ 00, when 00 dc‐gain Dongjun Lee Bode Plot Example 16 16 , sin 1. 10/ M 14.5 0.2, 173 2. 1/ 0 1,0 3. 4/ 12 4,90 4. 0/ 0 1 , 0 Dongjun Lee 3 Simple Bode Plot Examples 1. → → | | , ∠ 0 ‐20dB/decade roll‐off: low‐freq. attenuation 2. → → → 20log 20 log ∠ +20dB/decade: noisy high‐freq. amplification 3. → → w → 20log 20 log Im ∠ phase‐lead? ∠ acausal system Re Dongjun Lee Bode Plot of 1st Order System → ,∠ tan Bode Plot ‐ gain: | | 20 log 20log 10log 0 [dB] ‐ phase: tan 0 [rad] ‐ asymptotes (i) ≪: gain 0[dB], phase 0 roll‐off: ‐20dB/decade (ii) ≫: gain 20log 20log , phase 90 (iii) : gain 20log 23[dB], phase 45 bandwidth : cut‐off frequency Dongjun Lee 4 Bode Plot of 2nd Order System w/ _ → ∠ tan 0 Bode Plot ‐ asymptotes (i) ≪ : gain 0[dB], phase 0 roll‐off: ‐40dB/decade (ii) ≫ : gain 40log 40log , phase 180 (iii) : gain 20 log 0[dB] if 0.5, phase 90 no resonance if ‐ resonance (i.e., peak of | |: w 12 0 12, if 0.707 recall 1 and Dongjun Lee Bode Plot of 2nd Order System w/ polar plot →∞, 0 : gain 20 log 0[dB] if 0.5, phase 90 12 , if 0.707 Dongjun Lee 5 Additive Property of Bode Plot … 20 log 20log | | 20 log | | … 20 log | | ∠ ∠ ∠ 2 ⋯ ∠ ⋯ 1/ 20 log | | 20 log | |, ∠ ∠ 1/ Dongjun Lee Bode Plot of 2nd Order System w/ 1 1 1 input output sin → sin ∠ → sin ∠ ∠ 20 log || 20 log ||20log || ∠ ∠ ∠ Dongjun Lee 6 Example 7.3 1 16 16 ‐ low frequency: behaves like an integrator ‐ high frequency: behaves like a standard 2nd order system Dongjun Lee Base Motion & Transimissibility how much system dynamics oscillates? displacement transmissibility body how much forcing? motion force transmissibility chassis force ex) 1, , ⋅ peak 7.16dB [email protected]/s approx: | | 5 2.24 Dongjun Lee 7 Base Motion & Transimissibility worst‐case amplification: dc‐gain=1 vehicle speed 0.01 attenuation damping dc‐gain=0 Dongjun Lee Rotating Unbalance system dynamics sin vibration displacement sin → | | sin ∠ : eccentricity : unbalance mass e.g.: lathe, rotor, etc… transmitted force sin → | |sinwt ∠ Dongjun Lee 8 Rotating Unbalance sin what is and given rotation speed ? vibration displacement →| | sin ∠ transmitted force →| |sin ∠ ‐ bode plot of and to take into account ‐ effect of damping ? ‐ problematic if is small, yet, Dongjun Lee absorber.m rotate_m_u_absorber.m Dynamic Vibration Absorber , sin with zero at / sin ‐ if we set , s.t. / , →0even if ‐ zero vibration at / ; two extra peaks nearby though ‐ at , → force cancelation! Dongjun Lee 9 Seismograph system dynamics: 2 : direct measurement /2: damping top & bottom want to measure earthquake from ‐ if ≫ ‐ want to have a small large & soft (i.e., 0) Dongjun Lee Accelerometer system dynamics: 1 2 : direct measurement /2: damping top & bottom want to measure from ‐ if ≪ ‐ want to have a high small & stiff Dongjun Lee 10 More Comments on Bode Plots ‐ phase‐lead isn’t forecasting of a future: merely a phase‐shift in the sinusoid output in steady‐state sin ∠ ‐ is improper if lim → ∞ (i.e., o(N(s))>o(D(s))) → not realizable since it requires differentiation ‐ we may define Bode plot purely mathematically by 20log , ∠ : this then represents steady‐state gain/phase if is stable ‐ Bode gain‐phase relation: | | uniqely determines “minimum” ∠ | | ∠ ln slop sampling around * can’t design | | and ∠ independent * non‐minimum phase: | |, but non‐minimum ∠ Dongjun Lee Example: Non‐Minimum Phase System non_minimum.m 10 610 * same gain with , but non‐minimum phase shift * with RHP zeros non‐minimum phase phase‐shift is not minimum among the systems with the same gain plot * without RHP zeros (& delay) minimum‐phase system Dongjun Lee 11 Next Lecture ‐ control Dongjun Lee 12.