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 ∠ 0 0 , when 0 0 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 2 3[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 1 2 0
1 2 , 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
1 2 , 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 2.28@ 1.87rad/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 → | |sin wt ∠
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 6 10
* 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