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Unformatted text preview: 21 Chapter 2
Understanding Mechanical
System Requirements Exit 2001 by N. Mohan Print TOC " ! 22 Motivation
❏ How can the ASD accelerate and decelerate the load to
give desired speed profile Load ASD
ωL ω L ( rad / sec ) desired speed profile 100 0
Exit 2001 by N. Mohan 1 2 3 4 5 6 7 t (sec )
TOC " ! 23 Systems With Linear Motion
fe fL M x u= ⇒
dx
;
dt a= du
f − fL
= e
dt
M fM M u= dx
;
dt a= du
f
= M
dt
M x Figure on left includes load force, f L , that must be overcome
x Figure on right shows only the force, f M , available to accelerate
the mass, M
Accelaration
f −f
f
a= e L = M
M
M Exit 2001 by N. Mohan Power Input
Pe (t ) = f e ⋅ u = f M ⋅ u + f L ⋅ u Kinetic energy 1
WM = Mu 2
2 TOC " ! 24 Rotating Systems
f f
90 o β M r
torque θ θ Mg x Torque = force radius [ Nm ] [N] [ m] x Example: what torque is needed to hold M motionless
Exit 2001 by N. Mohan TOC " ! 25 ❏ Torque in an electric drive
ω
Motor Tem TL Load x Tem electromagnetic torque produced by motor
x Tem is opposed by load torque, TL
x The difference, Tem − TL = TJ , will accelerate the system
x d ω Tem − TL TJ
=
=
dt
J
J where J is the moment of inertia
Exit 2001 by N. Mohan TOC " ! 26 Calculation of Moment of Inertia J
of a Uniform Cylinder
! df
r rdθ
dM
dθ ω r1 d f = dM θ
dM = ρ rdθ dr
& & d
v
dt ⇒ dT = r 2 dM d
d
ω = ρ ( r 3 dr dθ d ! ) ω
dt
dt
2π ! 0 T = ρ ( ∫ r dr ∫ dθ ∫ d ! )
0 J solid =
Exit 2001 by N. Mohan 3 d!
& arc height length 0 r1 d! dr d
d
π
ω = ( ρ ! r14 ) ω
dt
2
" $# dt
# % 1
π
ρ ! r14 = M r12
2
2 J TOC " ! 27 Accelaration, Speed and Position, Power and
Energy
ωm
Motor Tem TL Tem + Load Σ
− TJ 1 α
J eq ∫ ωm ∫ θ TL acceleration ,
⇒ speed , dω m
TJ
1
α =
=
(Tem −TL ) =
dt
J eq
( J m+ J L ) ω m (t ) = ω m (0 ) + ∫0 α (τ ) dτ
t ⇒ position , θ (t ) = θ (0 ) + ∫0 ω (τ ) dτ
t Pem = Tem ⋅ω m ;
1
Kinetic Energy W = J ω 2
2
Power Exit 2001 by N. Mohan PL = TL ⋅ω m TOC " ! 28 Frictional Torque
Tf stiction 0 coloumb
friction T f = Bω
viscous friction stiction ω coloumb
friction x
x
x
x
Exit 2001 by N. Mohan Stiction: static component
Coulomb friction: dynamic component (constant magnitude)
Viscous friction: speed dependent
In general, friction is nonlinear
TOC " ! 29 ❏ Example: Aerodynamic drag
Drag power at different speeds
f L = 0.046 Cw Av 2 ; (Cw : drag coefficient) p = f L ⋅u
∴ power α speed 3
Speed Power (W) (km/h)
Cw = 0.3 Cw = 0.5 50
100 2001 by N. Mohan 1.44 kW 6.9 kW 150 Exit 0.86 kW
23.3 kW 11.5 kW
38.8 kW TOC " ! 210 Torsional Resonances
ωm ωL
TL Tem Motor Tshaft Load
JL d ωm
dt
d ωL
At load end Tshaft = TL + J L
dt
Tshaft
(θ m − θ L ) =
K
θ m and θ L :angular rotation at the two ends of the shaft
Jm At motor end Tshaft = Tem − J m x If K → ∞ , θ m = θ L
( J M and J L can be treated as one inertial mass )
x Finite K may lead to resonances
Exit 2001 by N. Mohan TOC " ! 211 Mechanical  Electrical Analogy
• Torque
• Angular Velocity • Voltage • Angular Displacement • Flux Linkage • Moment of Inertia • Capacitance • Spring Constant • 1/Inductance • Damping Coefficient • 1/Resistance • Coupling Ratio Exit • Current • Transformer ratio 2001 by N. Mohan TOC " ! 212 Electrical Analogy of Motor & Load
ωm TL Tem Motor ωL JL Jm ωM
Tem TJM TJL Tshaft
JM ωm ωL 1/ K Load TL Tem TJ TL JL J eq = J M + J L Finite shaft stiffness Exit 2001 by N. Mohan Infinite shaft stiffness TOC " ! 213 Coupling Mechanisms
❏ Required when
x a (rotary) motor is driving a load which requires linear
(translational) motion
x motors prefer higher rotational speed than that required by
the load
x the axis of rotation needs to be changed
❏ Types
! Conveyor belts (belt and pulley)
! Rack and pinion or a leadscrew type of arrangement
! Gear mechanisms
Exit 2001 by N. Mohan TOC " ! 214 Conversion between Linear and Rotary
Systems
fL
M u M = mass of load r ωm Motor r = pulley radius Tem du
f = M
+ fL
dt
Jm
u = r ωm
2 dω m
T = r f =r M
+ rf L
dt
d ωm
dω
2
Tem =
Jm
+r M
+ r fL
" $#
# dt%
"##dt
$##
%
required to accelerate
motor Exit 2001 by N. Mohan J m = motor inertia due to load
TOC " ! 215 Gears
Tem
Motor
JM T1 r1 ωM
r2 T2 TL
Load ωL JL x Basic relationships: radius, speed, torque
Equal speeds at gear surfaces ⇒ r1 ω M = r2 ω L
Power transferred across gears ⇒ ω M T1 = ω L T2 ,
⇒ r1
T
ω
= L = 1
r2
T2
ωM d ωM ωM
d ωL & Tem − J M
= TL + J L dt % ω L # dt % "## $###
#
"# $##
T1
T2
increased, torque decreased ω L > ω M ; T2 < T1 ; r2 < r1 x Geared up: speed
x Geared down: speed decreased, torque increased
ω L < ω M ; T2 > T1 ; r2 > r1 Exit 2001 by N. Mohan TOC " ! 216 Gears (cont’d)
x Equivalent Inertia
Tem 2 ω L d ωm ω L = Jm + J L dt + ω TL ωm m "## $###
#
%
J eq ⇒ J eq ω = Jm + JL L ωm 2 r = Jm + JL 1 r2 2 x Optimum gear ratio (to minimize Tem )
2 Jm r = 1 ⋅ JL r2 opt. and (Tem )opt. = 2 J m Exit 2001 by N. Mohan ⇒ r1 =
r 2 opt. Jm
JL r dω L
d ωm
= 2J m 2 dt r1 opt. dt
TOC " ! 217 Types of Loads
Centrifugal loads
Fan Constant Torque loads Hoist Exit 2001 by N. Mohan TOC " ! 218 Types of Loads
Squared power loads Compressor Constant power loads Winder Exit 2001 by N. Mohan TOC " ! 219 FourQuadrant Operation
ωm
(2) Tem ωm = +
Tem = −
p=− ωm
Load Motor Power 2001 by N. Mohan ωm = +
Tem = +
p=+
Tem ωm = −
Tem = −
p=+
(3) Exit (1) ωm = −
Tem = +
p=−
(4) TOC " ! 220 Dynamic Operation ❏ How the operating point changes with time
❏ Important for High Performance Drives
❏ Speed change: rapid and without any oscillations
❏ Requires good controller design Exit 2001 by N. Mohan TOC " ! 221 Summary
" What are the MKS units for force, torque, linear velocity,
angular velocity, speed, and power?
" What is the relationship between force, torque, and power?
" Show that torque is the fundamental variable in controlling
speed and position.
" What is the kinetic energy stored in a moving mass and a
rotating inertia? Exit 2001 by N. Mohan TOC " ! 222 Summary
❏What is the mechanism for torsional resonances?
❏ What are the various types of coupling mechanisms?
❏ What is the optimum gear ratio to minimize the torque
required from the drive to accelerate a load?
❏ What are the torquespeed and the powerspeed profiles
for various types of loads? Exit 2001 by N. Mohan TOC " ! ...
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 Fall '06
 Scalzo

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