446.328 mechanical system analysis
TRANSCRIPT
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Dongjun Lee
446.328 Mechanical System Analysis
기계시스템해석
Dongjun Lee (이동준)
Department of Mechanical & Aerospace EngineeringSeoul National University
Dongjun Lee
Today
‐ RLC circuit
‐mathematical analogy
‐ impedance
‐ op‐amp
‐ op‐amp circuits
‐ dc motors
‐ step motors
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RLC Components
variable resistance: potentiometer
voltage source: can provide any amount
of current (cf. constant pressure pump)
resistor: dissipate energy similar to damping [Ω
Ohm’s law
position sensing!
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RLC Components
charge stored in capacitor
capacitor [F]: two conductors separated by
non‐conducting medium
V can’t change instantaneously
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inductor [H]: coupling between current‐flow and induced magnetic field
magnetic flux due to coiling
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,1
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Kirchhoff’s Laws
1. voltage law:
sum of voltage drops around
a closed‐loop is zero
Δ Δ Δ Δ 0
2. current law:
sum of currents at a junction is zero
0ex) equivalent capacitance
1 1 parallel
series
equivalent resistance
series
parallel
equivalent inductance
series
parallel
impedance
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Impedance
impedance
ex) impedance as a transfer function
force
velocity
admittance
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force
velocity
passive
circuit
system
+
‐
passive
circuit
system
incident wave
reflected wave
scatteringoperator
ex) series and parallel connections
series parallel
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Mathematical Analogy
analogous! spring
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dampingmassposition
1
velocity
mathematically equivalent can use same tool/analysis across different domains!
(e.g., transfer function, state‐space approach, …)
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2ndOrder Response of RLC Circuit
1→
1
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‐ natural frequency
‐ damping ratio
‐ response of with unit‐step 1?‐ for circuits, usually over‐damped (e.g., L = 1mH, R = 200 ‐> C < 100nF)
‐ state‐space 0 11/ / , ⋯ → ,
small
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Input and Output Impedance
input impedance output impedance* we want:
1. very large input impedance
no effect on the source circuit w/o change in
2. very low output impedance
no voltage drop in and no effect by the loading circuit
* if ≫ , we can consider
two systems decoupled from each other
can analyze them separately!
system I system II
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Input/Output Impedance ‐ Example
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11
1
1
series
parallel
11 1
if ≫
* input impedance: circuit impedance seen from input source to ground
* output impedance: circuit impedance seen from output source with input short‐circuited
measurement system
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Operational Amplifier
‐ ideal op‐amp
1) infinite input impedance ∞ → 02) infinite open‐loop gain ∞ → 3) zero output impedance 0 →output =
‐ difference amplifier large open‐loop gain: 10 ~10
‐ : ~ Ω, 0Ω
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OP‐Amp Circuits ‐ I
virtual ground
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voltage follower
OP‐Amp Circuits ‐ II
high input impedance w/ low output impedance!
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V1
V2
R4
adjustable gain
if , 1 2
instrumentation amplifier
OP‐Amp Circuits ‐ III
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OP‐Amp Circuits ‐ IV
differentiator
integrator1
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time constant
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first‐order LPF
OP‐Amp Circuits ‐V
second‐order LPF
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11
1 1
1
v1
1
i2
i1
i
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‐ input impedance 2 Ω‐ input offset voltage for 0 2‐ input offset current for 0 20‐ output short circuit current = max. current from 25‐ common mode rejection ratio:
10 10 ← /2
‐ slew‐rate .
‐ bandwidth: (w/ fb,
to avoid instability in high‐frequency w/ ‐180deg phase‐lag
‐maximum output _
Real OP‐Amp (LM741C)
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Magneto‐Mechanical Coupling
i [x]
conductor
1) magnetic‐field induces force 2) motion induces voltage drop
f = BLi [z]
f: magnetic‐field induced forcei: currentB: magnetic field flux densityL: conductor length
conductor
v: motion [x]
E: back emf (electromotive force)v: velocityB: magnetic field flux density L: conductor length
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Armature‐Controlled Motor
torque
modeling
# of colis armatureradius
back emf
linearvelocity
angularvelocity
armaturelength
unwind armature coil
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dc‐Motor Modeling
mechanical side electrical side
externaltorque
‐ shaft torque proportional to current
‐ no‐load max. shaft velocity proportional to voltage (w/ 0‐ 1(if in ] and in / ])
kT: torque constant ke = 1/kn: speed
constant
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dc‐Motor System TF
mechanical side electrical side
often small
: torque constant 1/ : speed
constant
1
transfer matrix
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Example: Tachometer
back emf
governingequation
(tachometer)
i
Vo
1‐st order with /if 1[rad/s] ‐> →63% steady‐state value at
0
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Example: Speaker
systemmodeling
m
k
c
x,f
speaker coil +diaphragm
electrical amplifier
diaphragm & coilmove together
in B
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Real dc‐Motor
2: rotor – permanent magnet 5: stator – winding8: graphite brush or metal brush 7: commutator
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Commutator and Brush
‐ commutator: sequentially alternate current w.r.t. motor rotation
‐ brush (graphite/metal): maintain contact with rotating rotor ‐> friction
‐ brushless motor: proximity sensor + circuitry w/ NO mechanical contact
‐> less friction/ higher performance
* armature motor
brush
commutator
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dc‐motor data sheet
motor diagram
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Gearbox
[shaft w] = [motor w] / N
[shaft T] = N x [motor T] x η
‐ dc motor: designed for high speed and low torque
‐ real task: low speed and high force
‐ gear ratio = N : 1
motor wm output shaft wo
speed reduction
torqueamplification
efficiency
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Gearbox
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Encoder
- incremental quadrature encoder
three tracks: A, B, Index (+ A-, B-, Index- for differential)index pulse: zero position, number of turns
resolution = counts per turn * 4
current input position output?
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Encoder
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Resolver
‐ analog sensor (i.e., infinite‐resolution), absolute over a single turn
‐ started from military applications, in harsht industrial environments
‐ hot, humid, dusty, oily, or mechanically demanding environment
‐ AC voltage (6 VAC to 60 VAC, 400 Hz to 10,000 Hz) applied to the rotating coil
(rotor) induces a voltage across the gap in the stationary coil (stator)
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Encoder vs Resolver
Four unique states of the A and B signals known as quadrature output: A high –B low, A high – B high, A low – B high, and A low – B low.Since these four detectable states occur for every line on the codewheel, the resolution of the graduated lines can be multiplied by four.The smallest resolvable angle is then ¼ the angle between the coded lines ( = 360 / 4 * n). A 10 bit encoder (1024 lines) can be used to resolve 12 bits (4096 counts).
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PWM
input?
PWM (pulse width modulation)
- torque command signal is converted into pulses- higher duty cycle (i.e. 100*t/T[%]) -> higher current - much less power consumption than linear amplifier
T: period
t: width
linear amplifier:larger V*I dissipation
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Motor Driver (Amplifier)
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Stepper Motors
variable reluctance
iron tooth core
permanent magnet hybrid
obsoletebetter
larger torque
24‐48 cpt
cheaper
best
largest torque
100‐400 cpt
expensive
permanentmagnet
polarity‐changingstator
tooth‐attracting
stator
tootheadpermanentmagnet:
focused magneticflux
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Stepper Motor Principle
‐ digital pulse to change stator polarity ‐> angle increment
‐ pulse rate ‐> angular rate
‐ simple to control + low power
‐ no feedback: may be slipped or exceeding from desired angle!