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Theory of Machines and Automatic ControlWinter 2019/2020
Lecturer: Sebastian Korczak, PhD Eng.
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 2
Lecture 5 cont.
Dynamics of planar mechanisms.
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 3
Inverse dynamics problem
Dynamics of planar mechanisms
Calculation of forces and torques that cause given motion of a mechanism(kinetostatics)
0. Mechanism and it's geometry, driving and operating forces/torques, displacement, velocity and acceleration functions are given.
1. Calculation of inertia forces and torques acting moving members of the mechanism.
2. Decomposition of the mechanism with reaction disclosure.
3. Write down vector sums of external forces, reactions and inertia forces (d'Alembert equations).
4. Solve the equations with graphical and/or analytical method.
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 4
Inverse dynamic problem – example
Dynamics of planar mechanisms
A
B
E
D
C1
C2
C3
Pω
Given:geometry, mass, center of a mass locations, mass moments of inertia for all mechanism members, constant angular velocity ω of a driven element, operating force P.
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 5
Inverse dynamic problem – example
Dynamics of planar mechanisms
A
B
E
D
C1
C2
C3
ω
velocity scheme vB
vC1
vC 2
vC3
vD
ω3
ω2
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 6
Inverse dynamic problem – example
Dynamics of planar mechanisms
A
B
E
D
C1
C2
C3
acceleration scheme
aC 1
aB
aC 3
aDaC 2
aC 1
ε2
ε3
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 7
Inverse dynamic problem – example
Dynamics of planar mechanisms
A
B
E
D
C1
C2
C3
inertial forces and torques
aC 1
aC 3
aC 2
aC 1
ε2
ε3
BC 1
BC 2
BC 3
M B 2
M B 3
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 8
Inverse dynamic problem – example
Dynamics of planar mechanisms
A
B
E
D
C1
C2
C3
inertial forces and torques
BC 1
BC 2
BC 3
M B 2
M B 3
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 9
Inverse dynamic problem – example
Dynamics of planar mechanisms
A
B
E
D
C1
C2
C3
inertial forces and torques + external and operating forces and torques
BC 1
BC 2
BC 3
M B 2
M B 3
PM R
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 10
Inverse dynamic problem – example
Dynamics of planar mechanisms
A
A
B
E
E
D
C1
C2
C3
Structural decomposition
BC 1
BC 2
BC 3
M B 2
M B 3
PM R
B
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 11
Inverse dynamic problem – example
Dynamics of planar mechanisms
A
A
B
E
E
D
C1
C2
C3
Unknownreaction forces
BC 1
BC 2
BC 3
M B 2
M B 3
PM R
B
RB 2
RB1
RA 0RA 1
RE 3 RE 0
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 12
Inverse dynamic problem – example
Dynamics of planar mechanisms
E
E
DC2
C3
BC 2
BC 3
M B 2
M B 3
P
B
RB 2
RE 3 RE 0
RB 2+BC 2+ BC3+ P+ RE3=0
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 13
Inverse dynamic problem – example
Dynamics of planar mechanisms
E
E
DC2
C3
BC 2
BC 3
M B 2
M B 3
P
B
RB 2t
RE 3t
RB 2n
+ RB 2t
+ BC2+BC 3+ P+ RE 3n
+ RE 3t
=0
RB 2n
RE 3n
RE 0n
RE 0t
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 14
Inverse dynamic problem – example
Dynamics of planar mechanisms
E
E
DC2
C3
BC 2
BC 3
M B 2
M B 3
P
B
RB 2t
RE 3t
RB 2n
+ RB 2t
+ BC2+BC 3+ P+ RE 3n
+ RE 3t
=0
RB 2n
RE 3n
RE 0n
RE 0t
part BD : ∑ M D : ... .
RB 2t
=.. .
part ED : ∑ M D : ... .
RE 3t
=.. .
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 15
Inverse dynamic problem – example
Dynamics of planar mechanisms
E
E
DC2
C3
BC 2
BC 3
M B 2
M B 3
P
B
RB 2t
RE 3t
RB 2n
+ RB 2t
+ BC2+BC 3+ P+ RE 3n
+ RE 3t
=0
RB 2n
RE 3n
RE 0n
RE 0t
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 16
Inverse dynamic problem – example
Dynamics of planar mechanisms
E
E
DC2
C3
BC 2
BC 3
M B 2
M B 3
P
B
RB 2t
RE 3t
RB 2n
+ RB 2t
+ BC2+BC 3+ P+ RE 3n
+ RE 3t
=0
RB 2n
RE 3n
RE 0n
RE 0t
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 17
Inverse dynamic problem – example
Dynamics of planar mechanisms
E
E
DC2
C3
BC 2
BC 3
M B 2
M B 3
P
BRB 2
RE 3 RE 0
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 18
Inverse dynamic problem – example
Dynamics of planar mechanisms
E
E
DC2
C3
BC 2
BC 3
M B 2
M B 3
P
BRB 2
RE 3 RE 0
A
A
B
C1
BC 1
M R
RB1
RA 0 RA 1
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 19
Inverse dynamic problem – example
Dynamics of planar mechanisms
A
A
B
C1
BC 1
M R
RB1
RA 0 RA 1
part AB : R A 1+ BC 1+ RB1=0
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 20
Inverse dynamic problem – example
Dynamics of planar mechanisms
A
A
B
C1
BC 1
M R
RB1
RA 0
RA 1
M R=.. .
∑ M A : ... .
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 21
Lecture 6
Machine dynamics.Reduction of masses and forces.
Machine equation of motion.
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 22
Overview
Machine dynamics
Machine – a tool containing one or more parts that uses energy to perform an intended action. Machines are assembled from components.
source: wikipedia.org, The Boulton & Watt Steam Engine, 1784
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 23
Overview
Machine dynamics
time
angu
lar
velo
city
starting steady-state motion
stopping
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 24
Idea of reduction
Reduction of masses and forces
n-DoFSystem
x1(t )=F1(x1 , x2 , ... , t )x2(t )=F2(x1 , x2 , ... , t )
...xn(t )=Fn(x1 , x2 , ... , t )+constraints+ limitations
complicated
Source: James Albert Bonsack (1859 – 1924) - U.S. patent 238,640cigarette rolling machine, invented in 1880 and patented in 1881
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 25
Idea of reduction
Reduction of masses and forces
n-DoFSystem
simpler, but not always possible
or
mr(t)F r(t )
xr( t)
I r(t)
M r(t)
φr (t)
1-DoFSystem
Source: James Albert Bonsack (1859 – 1924) - U.S. patent 238,640cigarette rolling machine, invented in 1880 and patented in 1881
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 26
Kinetic energy
Reduction of masses
Total kinetic energy
T=∑i=1
n
(12mi v i
2+
12I iωi
2)
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 27
Kinetic energy
Reduction of masses
mr(t)F r(t )
xr( t)
Total kinetic energy
T =12mr v r
2
reduced mass
v r=dxr (t )
dt
T=∑i=1
n
(12mi v i
2+
12I iωi
2)
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 28
Kinetic energy
Reduction of masses
mr(t)F r(t )
xr( t)
I r(t)
M r(t)
φr (t)
Total kinetic energy
T =12I r ωr
2
reduced moment of inertia
T =12mr v r
2
reduced mass
or
v r=dxr (t )
dt
ωr=d φr (t)
dt
T=∑i=1
n
(12mi v i
2+
12I iωi
2)
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 29
System power
Reduction of forces
Total system's power
P (F i , M i ,ωi , v i , ...)
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 30
System power
Reduction of forces
Total system's power
P (F i , M i ,ωi , v i , ...)
P=Fr vr
reduced force
mr(t)F r(t )
xr( t)
v r=dxr (t )
dt
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 31
System power
Reduction of forces
Total system's power
P=M rωr
reduced torque
P (F i , M i ,ωi , v i , ...)
P=Fr vr
reduced force
or
mr(t)F r(t )
xr( t)
I r(t)
M r(t)
φr (t)
v r=dxr (t )
dt
ωr=d φr (t)
dt
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 32
Kinetic energy
Reduction of masses – details
T=∑i=1
n12mi v i
2+∑j=1
k12I jω j
2
n – translating elementsk – rotating elements
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 33
Kinetic energy
Reduction of masses – details
T=∑i=1
n12mi v i
2+∑j=1
k12I jω j
2
n – translating elementsk – rotating elements
12mr vr
2=∑
i=1
n12mi vi
2+∑j=1
k12I jω j
2
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 34
Kinetic energy
Reduction of masses – details
T=∑i=1
n12mi v i
2+∑j=1
k12I jω j
2
n – translating elementsk – rotating elements
12mr vr
2=∑
i=1
n12mi vi
2+∑j=1
k12I jω j
2
mr=∑i=1
n
mi
v i2
vr2+∑j=1
k
I jω j
2
v r2
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 35
Kinetic energy
Reduction of masses – details
T=∑i=1
n12mi v i
2+∑j=1
k12I jω j
2
n – translating elementsk – rotating elements
12mr vr
2=∑
i=1
n12mi vi
2+∑j=1
k12I jω j
2
mr=∑i=1
n
mi
v i2
vr2+∑j=1
k
I jω j
2
v r2
12I rωr
2=∑
i=1
n12mi v i
2+∑j=1
k12I jω j
2
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 36
Kinetic energy
Reduction of masses – details
T=∑i=1
n12mi v i
2+∑j=1
k12I jω j
2
n – translating elementsk – rotating elements
12mr vr
2=∑
i=1
n12mi vi
2+∑j=1
k12I jω j
2
mr=∑i=1
n
mi
v i2
vr2+∑j=1
k
I jω j
2
v r2
12I rωr
2=∑
i=1
n12mi v i
2+∑j=1
k12I jω j
2
I r=∑i=1
n
mi
vi2
ωr2+∑j=1
k
I jω j
2
ωr2
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 37
Kinetic energy
Reduction of masses – details
T=∑i=1
n12mi v i
2+∑j=1
k12I jω j
2
n – translating elementsk – rotating elements
12mr vr
2=∑
i=1
n12mi vi
2+∑j=1
k12I jω j
2
mr=∑i=1
n
mi
v i2
vr2+∑j=1
k
I jω j
2
v r2
12I rωr
2=∑
i=1
n12mi v i
2+∑j=1
k12I jω j
2
I r=∑i=1
n
mi
vi2
ωr2+∑j=1
k
I jω j
2
ωr2
– arbitrary chosen velocitiesvr , ωr
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 38
Work
Reduction of forces – details
dW=∑i=1
n
Pidsi cosαi+∑j=1
k
M jd φ j
n – translating elementsk – rotating elements
αi=∢(Pi , dsi)
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 39
Work
Reduction of forces – details
dW=∑i=1
n
Pidsi cosαi+∑j=1
k
M jd φ j
n – translating elementsk – rotating elements
Pr dsr=∑i=1
n
Pidsicosα i+∑j=1
k
M jd φ j
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 40
Work
Reduction of forces – details
dW=∑i=1
n
Pidsi cosαi+∑j=1
k
M jd φ j
n – translating elementsk – rotating elements
Pr dsr=∑i=1
n
Pidsicosα i+∑j=1
k
M jd φ j
Pr=∑i=1
n
Pidsidsr
cosα i+∑j=1
k
M j
d φ j
dsr
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 41
Work
Reduction of forces – details
dW=∑i=1
n
Pidsi cosαi+∑j=1
k
M jd φ j
n – translating elementsk – rotating elements
Pr dsr=∑i=1
n
Pidsicosα i+∑j=1
k
M jd φ j
Pr=∑i=1
n
Pidsidsr
cosα i+∑j=1
k
M j
d φ j
dsr
Pr=∑i=1
n
Piv idt
v rdtcosα i+∑
j=1
k
M j
ω jdt
vr dt
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 42
Work
Reduction of forces – details
dW=∑i=1
n
Pidsi cosαi+∑j=1
k
M jd φ j
n – translating elementsk – rotating elements
Pr dsr=∑i=1
n
Pidsicosα i+∑j=1
k
M jd φ j
Pr=∑i=1
n
Pidsidsr
cosα i+∑j=1
k
M j
d φ j
dsr
Pr=∑i=1
n
Piv idt
v rdtcosα i+∑
j=1
k
M j
ω jdt
vr dt
Pr=∑i=1
n
Piv iv r
cosαi+∑j=1
k
M j
ω j
vr
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 43
Work
Reduction of forces – details
dW=∑i=1
n
Pidsi cosαi+∑j=1
k
M jd φ j
n – translating elementsk – rotating elements
Pr dsr=∑i=1
n
Pidsicosα i+∑j=1
k
M jd φ j M r dφr=∑i=1
n
Pidsicosα i+∑j=1
k
M jd φ j
Pr=∑i=1
n
Pidsidsr
cosα i+∑j=1
k
M j
d φ j
dsr
Pr=∑i=1
n
Piv idt
v rdtcosα i+∑
j=1
k
M j
ω jdt
vr dt
Pr=∑i=1
n
Piv iv r
cosαi+∑j=1
k
M j
ω j
vr
M r=∑i=1
n
Pidsid φr
cosαi+∑j=1
k
M j
d φ j
d φr
M r=∑i=1
n
Pividt
ωrdtcosα i+∑
j=1
k
M j
ω jdt
ωrdt
M r=∑i=1
n
Piviωr
cosαi+∑j=1
k
M j
ω jωr
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 44
Reduction of forces/torques
Pr=∑i=1
n
Piv ivr
cosαi+∑j=1
k
M j
ω j
v rM r=∑
i=1
n
Piviωr
cosαi+∑j=1
k
M j
ω jωr
Reduction of masses/moments of inertia
mr=∑i=1
n
mi
v i2
v r2+∑j=1
k
I jω j
2
v r2 I r=∑
i=1
n
mi
vi2
ωr2+∑j=1
k
I jω j
2
ωr2
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 45
Linear motion
Machine equation of motion
m(t)F (t)
v (t)
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 46
Linear motion
Machine equation of motion
dT=dW
d (12m(t) v (t)2)=F ( t)dx
12dm(t )v (t)2
+m( t)v ( t)dv (t)=F (t)dx
12dm( t)v ( t)2
+m(t )dx (t )dt
dv (t )=F (t )dx
dm(t)dx
v (t)2
2+m
dv (t)dt
=F (t )
dm(t )dt
v (t )2
+mdv (t )dt
=F (t)
if m=const . ⇒ mdv (t)dt
=P(t) o r m x ( t )=F (t )
m(t)F (t)
v (t)
complete differential of kinetic energy
elementary work
elementary change of kinetic
energy
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 47
Angular motion
Machine equation of motion
dT=dW
d (I ω(t)2
2 )=M ( t)d φ
...
...dI (t)d φ
ω(t)2
2+ I (t)
dω(t )dt
=M (t )
dI ( t)dt
ω( t)2
+ I ( t)dω(t)dt
=M ( t)
if I=const . ⇒ Idω( t)dt
=M (t ) o r I φ(t)=M (t )
I (t)
M (t)
φ( t)
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 48
Rolling wheel (without a slip)
Reduction of masses and forces
Given: m – wheel's mass,IO – wheel's mass moment of inertia in point O,
r – wheel's radius,M – torque.
O
M
r
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Rolling wheel (without a slip)
Reduction of masses and forces
Given: m – wheel's mass,IO – wheel's mass moment of inertia in point O,
r – wheel's radius,M – torque.
O v
Assume: v – velocity of the wheel's center,ω – angular velocity of the wheel.
M
rω
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Rolling wheel (without a slip)
Reduction of masses and forces
Given: m – wheel's mass,IO – wheel's mass moment of inertia in point O,
r – wheel's radius,M – torque.
O v
Assume: v – velocity of the wheel's center,ω – angular velocity of the wheel.
M
rω
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 51
Rolling wheel (without a slip)
Reduction of masses and forces
Given: m – wheel's mass,IO – wheel's mass moment of inertia in point O,
r – wheel's radius,M – torque.
O v
Assume: v – velocity of the wheel's center,ω – angular velocity of the wheel.
M
rω
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 52
Rolling wheel (without a slip)
Reduction of masses and forces
Given: m – wheel's mass,IO – wheel's mass moment of inertia in point O,
r – wheel's radius,M – torque.
O v
T=12mv2
+12IOω
2 but v=ω r
T=12mv2
+12IOv2
r2 =12 (m+
IOr2 )v2
=12mr v
2
mr=m+IOr2 =const.
M
r
P=Mω
P=Mvr=Mrv=F r v
F r=Mr
mrdvdt
=F r
Assume: v – velocity of the wheel's center,ω – angular velocity of the wheel.
ω
(m+I 0
r2 ) dvdt =Mr
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 53
m1 – total mass
mr1 – reduced mass
m2 – total mass
mr2 – reduced mass
m1 = m
2
mr1 ? m
r2
Reduction of masses and forces – quiz
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 54
Example 1
Reduction of masses and forces
In this example a drum winch is analyzed. It consist of:● electric motor EM, which generates torque as a function of angular
velocity ω: M=A-Bω where A and B are given constants; rotor moment of
inertia is equal to Im
;
● two stage gearbox GB (reducer) with given gears moments of inertia I1, I
2,
I3, I
4; shafts moments of inertia are equal to I
s; gears ratio are given as
i1=ω
2 /ω
1 and i
2=ω
4 /ω
3
● winch's drum has diameter D and moment of inertia Id; drum is set on two
ball bearing that generates resistance Mf assumed to be constant;
● inclined plane with angle α related to the horizon line;● box of weight G pulled up with winch. Friction between object and inclined
plane is represented as dry friction with μ coefficient.
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Example 1
Reduction of masses and forces
M
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M
Example 1
Reduction of masses and forces
V
ω3
ω3
ω2
ω1
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Example 1
Reduction of masses and forces
V
ω3
ω3
ω2
ω1
T=12( Im+ I1+ I s)ω1
2+
12( I2+ I3+I s)ω2
2+
12( I 4+ Id+I s)ω3
2+
12Ggv2
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 58
Example 1
Reduction of masses and forces
V
ω3
ω3
ω2
ω1
T=12( Im+ I1+ I s)ω1
2+
12( I2+ I3+I s)ω2
2+
12( I 4+ Id+I s)ω3
2+
12Ggv2
ω2ω1
=i1
ω3ω2
=i2
v=D2
ω3
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 59
Example 1
Reduction of masses and forces
V
ω3
ω3
ω2
ω1
T=12( Im+ I1+ I s)ω1
2+
12( I2+ I3+I s)ω2
2+
12( I 4+ Id+I s)ω3
2+
12Ggv2
ω2ω1
=i1→ω2=ω1 i1
ω3ω2
=i2→ω3=ω2i2=ω1i1i2
v=D2
ω3=D2
ω1i1i2
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Example 1
Reduction of masses and forces
V
ω3
ω3
ω2
ω1
T=12( Im+ I1+ I s)ω1
2+
12( I2+ I3+I s)ω1
2 i12+
12( I4+ Id+ I s)ω1
2 i12 i2
2+
12GgD2
4ω1
2 i12 i2
2
ω2ω1
=i1→ω2=ω1 i1
ω3ω2
=i2→ω3=ω2i2=ω1i1i2
v=D2
ω3=D2
ω1i1i2
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12.11.2019 TM&AC, Lecture 5, Sebastian Korczak, only for educational purposes 61
Example 1
Reduction of masses and forces
V
ω3
ω3
ω2
ω1
T=12 [( Im+ I1+ I s)+(I 2+I 3+ I s)i1
2+( I 4+ Id+I s)i1
2 i22+GgD2
4i1
2 i22]ω1
2
ω2ω1
=i1→ω2=ω1 i1
ω3ω2
=i2→ω3=ω2i2=ω1i1i2
v=D2
ω3=D2
ω1i1i2
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Example 1
Reduction of masses and forces
V
ω3
ω3
ω2
ω1
I R=Im+ I 1+ I s+( I 2+ I 3+ I s)i12+( I 4+ I d+ I s)i1
2i22+GgD2
4i1
2 i22
ω2ω1
=i1→ω2=ω1 i1
ω3ω2
=i2→ω3=ω2i2=ω1i1i2
v=D2
ω3=D2
ω1i1i2
reduced moment of inertia
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M
Example 1
Reduction of masses and forces
V
ω3
ω3
ω2
ω1
P
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M
Example 1
Reduction of masses and forces
V
ω3
ω3
ω2
ω1
N=M sω1−M f ω3−P v
P
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Reduction of masses and forcesExample 1
P
α
P=…
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Reduction of masses and forcesExample 1
P
α
P=…
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Reduction of masses and forcesExample 1
P
α
P=…
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Reduction of masses and forcesExample 1
P
α
P=T+G sinα=μN+G sinα=μG cosα+G sinα
GTN
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M
Example 1
Reduction of masses and forces
V
ω3
ω3
ω2
ω1
N=M sω1−M f ω3−(μGcosα+G sinα) v
P
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M
Example 1
Reduction of masses and forces
V
ω3
ω3
ω2
ω1
N=M sω1−M f ω1 i1i2−(μGcosα+G sin α)D2
ω1i1i2
P
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M
Example 1
Reduction of masses and forces
V
ω3
ω3
ω2
ω1
N=[M s−M f i1i2−(μGcosα+G sinα)D2i1i2]ω1
P
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M
Example 1
Reduction of masses and forces
V
ω3
ω3
ω2
ω1
M R=M s−M f i1 i2−(μG cosα+G sinα)D2i1i2
P
reduced torque
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Example 1
Reduction of masses and forces
M R=M s −(M f i1 i2+(μGcosα+G sinα)D2i1i2)
Reduced torque
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Example 1
Reduction of masses and forces
M R=M s −(M f i1 i2+(μGcosα+G sinα)D2i1i2)
Reduced torque
driving torque MD(electric motor torque)
passive torque MP
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Example 1
Reduction of masses and forces
Reduced torque
M R = M D−M P = (A−Bω1)−MP
M R=M s −(M f i1 i2+(μGcosα+G sinα)D2i1i2)
driving torque MD(electric motor torque)
passive torque MP
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Reduction of masses and forcesExample 1
I Rdω1
dt=M R
IR (t)
MR (t)
ω1(t )
M R=A−Bω1−M P
Start-up process
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Reduction of masses and forcesExample 1
I Rdω1
dt=M R
IR (t)
MR (t)
ω1(t )
M R=A−Bω1−M P
Start-up process
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Reduction of masses and forcesExample 1
I Rdω1
dt=M R
IR (t)
MR (t)
ω1(t )
M R=A−Bω1−M P
Start-up process
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Reduction of masses and forcesExample 1
Start-up process
dω1
dt+BI R
ω1=A−M P
I RA ,M P ,B , I R - constants
general solution particular solution
non-homogeneous 1st order ODE
with const. coef.
initial condition
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Reduction of masses and forcesExample 1
Start-up process
dω1
dt+BI R
ω1=A−M P
I RA ,M P ,B , I R - constants
general solution particular solution
non-homogeneous 1st order ODE
with const. coef.
initial condition
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Reduction of masses and forcesExample 1
Start-up process
dω1
dt+BI R
ω1=A−M P
I RA ,M P ,B , I R - constants
general solution particular solution
non-homogeneous 1st order ODE
with const. coef.
ω1g(t)=Ee−BIRt
ω1 p(t)=F
initial condition
ω1(t=0)=0
ω1(t)=A−M P
B(1−e
−BI Rt)
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Reduction of masses and forcesExample 1
Start-up process
ω1(t)=A−M P
B(1−e
−BI Rt)
t
ω1(t )
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Reduction of masses and forcesExample 1
Start-up process
ωmax=A−M P
B
t
ω1(t )
steady state value (maximum)
ω1(t)=A−M P
B(1−e
−BI Rt)
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Reduction of masses and forcesExample 1
Start-up process
ω1(t)=A−M P
B(1−e
−BI Rt)
0,95A−M P
B=A−M P
B(1−e
−BI Rt95)
ωmax=A−M P
B
t
ω1(t )
steady state value (maximum)
START-UP TIME(95% of maximum value)
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Reduction of masses and forcesExample 1
Start-up process
ω1(t)=A−M P
B(1−e
−BI Rt)
ωmax=A−M P
B
t
ω1(t )
steady state value (maximum)
START-UP TIME(95% of maximum value)
t 95≈3I RB
0,95A−M P
B=A−M P
B(1−e
−BI Rt95)
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Reduction of masses and forcesExample 1
Start-up process
ω1(t)=A−M P
B(1−e
−BI Rt)
v(t)=D2
ω1(t)i1 i2
electric motor angular velocity
box linear velocity
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Reduction of masses and forcesExample 2
air resistance proportional to velocity
I sI 1
I 2
I 3
Iwm
r
MF
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Reduction of masses and forcesExample 2
I sI 1
I 2
I 3
Iwm
r
MF
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Reduction of masses and forcesExample 2
I sI 1
I 2
I 3
Iwm
r
MF
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Reduction of masses and forcesExample 2
I sI 1
I 2
I 3
Iwm
r
MF
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Reduction of masses and forcesExample 2
I sI 1
I 2
I 3
Iwm
r
MF