Dongjun Lee
M2794.25 Mechanical System Analysis 기계시스템해석
‐ lecture 2,3,4,5 ‐
Dongjun Lee (이동준)
Department of Mechanical & Aerospace Engineering Seoul National University
Dongjun Lee
Lectures 2‐5
Laplace transform
‐ partial fraction expansion
‐ characteristic roots
‐ final value theorem (FVT)
‐ transfer function
‐ equilibrium
‐ impact and IVT
‐ effect of zero
‐ current reading: Ch.3; next Ch. 4
* mid‐term exam 4/19 (W) 7:00‐9:00pm
* final‐term exam 6/14 (W) 7:00‐9:00pm
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LTI System Dynamics
mechanical systems
electrical systems
fluid systems
LTI differential equation:
unified system behavior?
* LTI: linear time‐invariant (linear and all system parameters constants)
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s‐Domain and t‐Domain
(complex) s‐
need to time‐integrate…
simply multiply!
impulse response
transfer function
u(t) u(t)
U(s) U(s)
causal h
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Laplace Transform
‐typically 0for 0 ( : right‐sided signal)
‐ : complex number
‐transform a signal from t‐domain to s‐domain definition
existence (i.e., integration exists)
‐exists, if is of exponential order: ∃ s.t. lim→ 0 region of attraction (ROC)
‐existence depends on
‐ROC extends to right
if
∈
Re Im
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ROC and Stability
‐although will be taught later with more details:
t 0
impulse response
∗ convolution
‐BIBO (bounded‐input bounded‐output) stability:
is stable, if is bounded ∀ 0and for allbounded
| | | | should be bounded
‐in fact, is stable iff
1. should have | 0 2. can’t have poles in RHP (e.g., 1/(s‐1))
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Essential Examples
for more, see the Tables in the book
unit step ramp
exponential* sine cosine
linearity
ex: pulse 1 1 1
unit impulse lim
→ / 1
time‐shift
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Laplace Transform Properties
differentiation*
integration*
time‐shift s‐shift
ramp‐modulation
sin cos
e.g.: damped oscillation
for more, see the Tables in the book
decaying frequency
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Partial Fraction Expansion
partial fraction expansion
…
⋯ ⋯
1
1 !
thus,
characteristic roots of | 0 for this case, ,
Re
Im 1) if a>0 & p>0 characteristic roots in LHP
→ 0 sin → 0, → 0 2) if a<0 || p<0 characteristic roots in RHP
→ ∞ sin → ∞, → ∞ 3) if a=0, p=0 and simplecharacteristic roots in jw‐axis
x(t) has sustained oscillation/offset α cos , xxx ‐a
x
x
+bj
‐bj
‐p
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Example 1
3 5 , 0 10
10 5
3 5 9
5 9
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Example 2
2 5 5 , 0 0 0
2 1
2 2
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Characteristic Roots
characteristic roots of | 0
1) if a>0 & p>0 characteristic roots in LHP
→ 0 sin → 0, → 0 2) if a<0 || p<0 characteristic roots in RHP
→ ∞ sin → ∞, → ∞ 3) if a=0, p=0 and simplecharacteristic roots in jw‐axis
x(t) has sustained oscillation/offset α cos ,
Re Im
xxx ‐a x
x
+bj
‐bj
‐p sin
sin , cos ,
exponential
decay frequency [rad/s]
partial fraction expansion
… ⋯ ⋯
1
1 !
4) if a=0, p=0 but not simplerepeated roots on jw‐axis
→ ∞ α cos ,
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Example
, 0 1, 0 0, /2
mass‐spring‐damper system
1 0 0 0 1
free response: by initial condition w/ zero input
forced response: by input w/ zero initial condition
characteristic roots: 0,
1) if 0 0, / oscillation with offset
2) if 4 0, decaying oscillation + offset
3) if 4 0, no‐oscillation + offset
4) if 0 0, → ∞
Re Im
xxx ‐a
x
x +bj
‐bj
‐p
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Final Value Theorem
‐ can we know ∞ without solving partial fraction?
final value theorem:
if characteristic roots of are all in LHP or with a simple root at 0
* caution!: FVT not applicable if has charact. roots in RHP or on ‐axis;
or multiple roots at s=0
‐example: mass‐spring‐damper system
? ?
lim
→lim
→
‐(incomplete) proof
lim→ ∞ 0 lim
→ 0
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LTI System Response
, 0 1, 0 0, 1
mass‐spring‐damper system
1 0 0 0 1
free response: by initial condition w/ zero input
forced response: by input w/ zero initial condition
1
transient: vanishing steady‐state:
sustained
* LTI system response
= free response (initial condition) + forced response (input)
= transient response (vanishing) + steady‐state response
transient vanishing if all characteristic roots from H(s) are in LHP
linear!
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Example
equation of motion
solution
free
response forced response
steady‐state (i.e. t∞)
transient (i.e. 0 as t∞) Laplace analysis
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Example
equation of motion
solution f = step input
free response
forced response Laplace analysis
free response + forced response both steady‐state: harmonic oscillation
with natural frequency / + offset
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Transfer Function
H(s)
input transfer function output
1 0 0 0 1
‐this H(s) contains all the information about system dynamics:
H(s) represents system dynamics in s‐domain! transfer function
or
‐transfer function representation is valid only: 1) with zero initial condition;
or 2) for steady‐state when system is stable transfer function (input‐to‐output)
just multiplication!
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Stability
transfer function system response
zeros | 0
poles | 0
1) is stable, if all poles of are in LHP:
→ 0for any initial condition (with zero input) bounded for any bounded input u(t)
2) is unstable, if any pole of is in RHP (or repeated poles on jw‐axis):
→ ∞for all initial conditions; → ∞for any bounded input 3) is marginally stable, if poles on jw‐axis are simple with all others in LHP:
will have constant offset/oscillation for all initial condition
→ ∞for some bounded input
Re Im
xxx ‐a x
x +bj
‐bj
‐p
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Example: Pendulum
sin 0
1) downward position with 0: stable
3) upward position: unstable
2) downward position with 0: marginally stable
4) equilibrium: if system starts there, it will stay there , 0,0 stable equilibrium , 0, unstable equilibrium
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Impulse Response via Mechanics
from mechanics k
b impact!
0 , 0
0 0
starts from t=0
1/ 1 1
affects directly, but takes some time to affect ,
, still bounded for 0,0
integration of , during [0,0 0
0 0
0 0 1
can we reach the same conclusion with Laplace transform?
impulse
* impulse initial velocity jumps!
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Impulse Response via Laplace Transform
initial value theorem
0 0
starts from t=0
0 0
0 lim
→ 0 0 1
0 , 0 lim
→ lim
→
if , exist; and lim
→ is computable
* applicable even for unstable systems
* here (i.e., from 0, not from 0 )
0 0 0 1
example:
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Zeros
In general, LTI system dynamics is given by
H(s)
input transfer function output
⋯ ⋯
n times differentiation
Then, the transfer function (i.e., with zero initial condition) is given by:
⋯
⋯
‐ is: proper if , strictly‐proper if , acausal if e.g., integrator: 1/ differentiator:
‐impulse response = : contains all information about system
‐poles 0 stability
‐zeros 0 effects?
ex)
ex) , 1
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Effects of Zeros
input output
transfer function
1) zeroing (or blocking effect) of input signals
Re Im
zx
For proper stable system ,
if , blocked even if is unbounded! in theory though...
if , sinusoid signal with [rad/s] (e.g., sin ) all filtered out!
2) inverse response
1
‐inverse response (or initial undershoot) if has odd number of positive real zeros
‐zero crossing if has a positive real zero z
‐ w/ RHP zeros close to 0or close to RHP poles difficult to control! (시스템제어이론)
0
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Next Lecture
‐ mechanical system modeling (Ch. 4)