Dongjun Lee
446.328 Mechanical System Analysis
기계시스템해석‐ lecture 15‐
Dongjun Lee (
이동준)
School of Mechanical & Aerospace Engineering Seoul National University
First Order System Examples
/
1
/
time constant
volume rate
/
density
resistance
Dongjun Lee
First Order Systems
first order system
(standard form) with
0
1
1 0 1
1
:time constant [sec]
smaller : faster response larger ∶slower response
free response
1
1/ 0 0
Re Im
1/x
0.368
0.0183
s‐domain
faster slower no oscillation
Dongjun Lee
First Order System: Step Response
1 , 0 0
1
1 0 1
1
step response1
]0.368 0.0183
FVT
transient & steady‐state
0
1
no overshoot
0.63
Dongjun Lee
First Order System: Ramp Response
, 0 0
1
1 0 1
1
step response‐> 1
smaller => faster response w/ less tracking error
FVT
lim
→First Order System: Parameter Estimation
/
,/
,lim
→lim
→/ 1
FVT
∞ /
IVT
0 /
c
m
Dongjun Lee
Second Order Systems
second order system
(standard form)
2
input natural freq.
damping ratio
/
,2 / ‐>
/
1/
,, ,
1.5
to make → in steady‐state
Dongjun Lee
Second Order System: Free Response
second order system
(standard form)
2
input natural freq.
damping ratio
1
2 0 0 2 0
2
free response forced response
Re Im
‐ x
x +
‐
2 0
characteristic equation:
1
1. 0(un‐damped):
2. 0 1(under‐damped): 1
≔ 1 : damped‐frequency
3. 1(critically‐damped): ‐> no oscillation
4. 0(over‐damped): 1
dominant root: 1
Dongjun Lee
Second Order System: Pole Locations
second order system
(standard form)
2
input natural freq.
damping ratio
Re Im
‐ x
x
+ 1
‐ 1
- : how fast is the convergence (
‐ 1 : damped oscillation frequency
‐ tan : depends only on
‐radius
Second Order System: Pole Locations
second order system (standard form)
2
input natural freq.
damping ratio
Re Im
‐ x
x
+ 1
‐ 1
Dongjun Lee
Second Order System: Step Response
2 2
1/ , 0 0 0
1. 0(un‐damped): 1 cos
2. 0 1(under‐damped): 1 cos sin
3. 1(critically‐damped): 1
4. 0(over‐damped): 2 second order system
(standard form)
2
input natural freq.
damping ratio
Dongjun Lee
Step Response of Under‐Damped Systems
second order system
(standard form)
2
2. 0 1(under‐damped): 1 cos sin
:maximum overshoot : peak time
: rise time : settling time : 50% delay time
Dongjun Lee
Step Response of Under‐Damped Systems
1 cos sin
1 1
1 sin
1 1
1 sin
:maximum overshoot : peak time
: rise time : settling time
: 50% delay time
0 tan 1
2 3
2
Maximum Overshoot & Peak Time
0 → tan tan → , 0,1, . .
1 4
1 1
1. c++ => t_p ++, M_p ‐‐
2. M_p depends only on : can be used to estimate 3. k++ => t_p ‐‐, M_p ++
4. m++ => M_p ++, t_p ++
1 1
1 sin
Dongjun Lee
Rise, Settling, and Delay Time
w_n ++ => t_r‐‐
100% rise time
1 → sin 0
1 1
1 sin
3 /2 2 → →
tan 1
2% settling time4 2
c ++ or m‐‐ => t_s‐‐
50% delay time
0.5
1 0.7 1 0.7
2 c ‐‐ or k++ => t_d++
Dongjun Lee
Second Order System: Effect of
second order system
(standard form)
2
Re Im
‐ x
x
+ 1
‐ 1
Dongjun Lee
Second Order System: Effect of Pole Locations
Second Order System: Parameter Estimation
/
, ,1
1 sin
→ …
…
1 ln , 2
Dongjun Lee
MatLab Example
first‐order system
second‐order system
with zero
,
→ , 0
/
, ,1
,
4 2
Dongjun Lee
Second Order System: Example
/
,2 / ‐>
1 4 /
/
1. changing c
Re Im
‐ x
x
+ 1
‐ 1
2. changing m
3. changing k
Dongjun Lee
Next Lecture
‐ fluid system modeling
