Post on 22-Aug-2020
TM&AC - Winter 2019/2020
velocities and accelerations in planar mechanismsEXAMPLE
Given: mechanism geometry andangular displacement φ(t)t)) of driven element.of of driven element.driven of driven element.element).
Warsaw University of Technology, The Faculty of Automotive and Construction Machinery EngineeringInstitute of Machine Design Fundamentals, Department of Mechanics, http://www.ipbm.simr.pw.edu.pl/,
Sebastian Korczak, PhD Eng.
φ(t)t)
A B
C
D
e
h
O
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 2
φ(t)t)
AB
C
D
e
h
1
23 4
5
O
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 3
φ(t)t)
AB
C
D
e
h
1
23 4
5
O
X
Y
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 4
φ(t)t)
AB
C
D
e
h
1
23 4
5
O
X
Y
r
ha
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 5
φ(t)t)
AB
C
D
e
h
1
23 4
5
O
X
Y
r
ha
c b
d
e
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 6
φ(t)t)
X
Y
r
ha
cb
d
e
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 7
φ(t)t)
X
Y
r
ha
cb
d
e
φh
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 8
φ(t)t)
X
Y
r
ha
cb
d
e
φhφ
h=90o
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 9
φ(t)t)
X
Y
r
h
a
cb
d
e
φh
φa
φh=90o
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 10
φ(t)t)
X
Y
r
h
a
cb
d
e
φh
φa(t)t)
φh=90o
φa(t)t)
φa(t)t) ≠ const.
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 11
φ(t)t)
X
Y
r
h
a
cb
d
e
φh
φa(t)t)
φh=90o
φa(t)t)
φa(t)t) ≠ const.
φr
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 12
φ(t)t)
X
Y
r
h
a
cb
d
e
φh
φa(t)t)
φh=90o
φa(t)t)
φa(t)t) ≠ const.
φr(t)t)
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 13
φ(t)t)
X
Y
r
h
a
cb
d
e
φh
φa(t)t)
φh=90o
φa(t)t)
φa(t)t) ≠ const.
φr(t)t)
φr(t)t) = 90o - φ(t)t)
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 14
φ(t)t)
X
Y
r
h
a
cb
d
e
φh
φa(t)t)
φh=90o
φa(t)t)
φa(t)t) ≠ const.
φr(t)t)
φr(t)t) = 90o - φ(t)t)
φb
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 15
φ(t)t)
X
Y
r
h
a
cb
d
e
φh
φa(t)t)
φh=90o
φa(t)t)
φa(t)t) ≠ const.
φr(t)t)
φr(t)t) = 90o - φ(t)t)
φb(t)t)
φb(t)t) ≠ const.
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 16
φ(t)t)
X
Y
r
h
a
cb
d
e
φh
φa(t)t)
φh=90o
φa(t)t) ≠ const.
φr(t)t)
φr(t)t) = 90o - φ(t)t)
φb(t)t)
φb(t)t) ≠ const.
φc(t)t)
φc(t)t) = φ
a(t)t)
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 17
φ(t)t)
X
Y
r
h
a
cb
d
e
φh
φa(t)t)
φh=90o
φa(t)t)
φa(t)t) ≠ const.
φr(t)t)
φr(t)t) = 90o - φ(t)t)
φb(t)t)
φb(t)t) ≠ const.
φc(t)t)
φc(t)t) = φ
a(t)t)
φd
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 18
φ(t)t)
X
Y
r
h
a
cb
d
e
φh
φa(t)t)
φh=90o
φa(t)t)
φa(t)t) ≠ const.
φr(t)t)
φr(t)t) = 90o - φ(t)t)
φb(t)t)
φb(t)t) ≠ const.
φc(t)t)
φc(t)t) = φ
a(t)t)
φd
φd=270o
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 19
φ(t)t)
X
Y
r
h
a
cb
d
e
φh
φa(t)t)
φh=90o
φa(t)t)
φa(t)t) ≠ const.
φr(t)t)
φr(t)t) = 90o - φ(t)t)
φb(t)t)
φb(t)t) ≠ const.
φc(t)t)
φc(t)t) = φ
a(t)t)
φd
φd=270o
φe
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 20
φ(t)t)
X
Y
r
h
a
cb
d
e
φh
φa(t)t)
φh=90o
φa(t)t)
φa(t)t) ≠ const.
φr(t)t)
φr(t)t) = 90o - φ(t)t)
φb(t)t)
φb(t)t) ≠ const.
φc(t)t)
φc(t)t) = φ
a(t)t)
φd
φd=270o
φeφ
e=180o
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 21
φ(t)t)
X
Y
r
h
a
cb
d
e
φh
φa(t)t)
φh=90o
φa(t)t)φ
a(t)t) ≠ const.
φr(t)t)
φr(t)t) = 90o - φ(t)t)
φb(t)t)
φb(t)t) ≠ const. φ
c(t)t)
φc(t)t) = φ
a(t)t)
φd
φd=270o
φe
φe=180o
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 22
φ(t)t)
X
Y
r
h
a
cb
d
e
φh
φa(t)t)
φh=90o
φa(t)t)φ
a(t)t) ≠ const.
φr(t)t)
φr(t)t) = 90o - φ(t)t)
φb(t)t)
φb(t)t) ≠ const. φ
c(t)t)
φc(t)t) = φ
a(t)t)
φd
φd=270o
φe
φe=180o
|r| = |AB| = r = const.
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 23
φ(t)t)
X
Y
r
h
a
cb
d
e
φh
φa(t)t)
φh=90o
φa(t)t)φ
a(t)t) ≠ const.
φr(t)t)
φr(t)t) = 90o - φ(t)t)
φb(t)t)
φb(t)t) ≠ const. φ
c(t)t)
φc(t)t) = φ
a(t)t)
φd
φd=270o
φe
φe=180o
|r| = |AB| = r = const.
|h| = |OA| = h = const.
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 24
φ(t)t)
X
Y
r
h
a
cb
d
e
φh
φa(t)t)
φh=90o
φa(t)t)φ
a(t)t) ≠ const.
φr(t)t)
φr(t)t) = 90o - φ(t)t)
φb(t)t)
φb(t)t) ≠ const. φ
c(t)t)
φc(t)t) = φ
a(t)t)
φd
φd=270o
φe
φe=180o
|r| = |AB| = r = const.
|h| = |OA| = h = const.
|a| = |OB| = a(t)t)
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 25
φ(t)t)
X
Y
r
h
a
cb
d
e
φh
φa(t)t)
φh=90o
φa(t)t)φ
a(t)t) ≠ const.
φr(t)t)
φr(t)t) = 90o - φ(t)t)
φb(t)t)
φb(t)t) ≠ const. φ
c(t)t)
φc(t)t) = φ
a(t)t)
φd
φd=270o
φe
φe=180o
|r| = |AB| = r = const.
|h| = |OA| = h = const.
|a| = |OB| = a(t)t)
|c| = |OC| = c = const.
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 26
φ(t)t)
X
Y
r
h
a
cb
d
e
φh
φa(t)t)
φh=90o
φa(t)t)φ
a(t)t) ≠ const.
φr(t)t)
φr(t)t) = 90o - φ(t)t)
φb(t)t)
φb(t)t) ≠ const. φ
c(t)t)
φc(t)t) = φ
a(t)t)
φd
φd=270o
φe
φe=180o
|r| = |AB| = r = const.
|h| = |OA| = h = const.
|a| = |OB| = a(t)t)
|c| = |OC| = c = const.
|b| = |CD| = b = const.
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 27
φ(t)t)
X
Y
r
h
a
cb
d
e
φh
φa(t)t)
φh=90o
φa(t)t)φ
a(t)t) ≠ const.
φr(t)t)
φr(t)t) = 90o - φ(t)t)
φb(t)t)
φb(t)t) ≠ const. φ
c(t)t)
φc(t)t) = φ
a(t)t)
φd
φd=270o
φe
φe=180o
|r| = |AB| = r = const.
|h| = |OA| = h = const.
|a| = |OB| = a(t)t)
|c| = |OC| = c = const.
|b| = |CD| = b = const.
|d| = d(t)t)
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 28
φ(t)t)
X
Y
r
h
a
cb
d
e
φh
φa(t)t)
φh=90o
φa(t)t)φ
a(t)t) ≠ const.
φr(t)t)
φr(t)t) = 90o - φ(t)t)
φb(t)t)
φb(t)t) ≠ const. φ
c(t)t)
φc(t)t) = φ
a(t)t)
φd
φd=270o
φe
φe=180o
|r| = |AB| = r = const.
|h| = |OA| = h = const.
|a| = |OB| = a(t)t)
|c| = |OC| = c = const.
|b| = |CD| = b = const.
|d| = d(t)t)
|e| = e = const.
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 29
φ(t)t)
X
Y
r
h
a
cb
d
e
φh
φa(t)t)
φh=90o
φa(t)t)φ
a(t)t) ≠ const.
φr(t)t)
φr(t)t) = 90o - φ(t)t)
φb(t)t)
φb(t)t) ≠ const. φ
c(t)t)
φc(t)t) = φ
a(t)t)
φd
φd=270o
φe
φe=180o
|r| = |AB| = r = const.
|h| = |OA| = h = const.
|a| = |OB| = a(t)t)
|c| = |OC| = c = const.
|b| = |CD| = b = const.
|d| = d(t)t)
|e| = e = const.
h + r = ac + b + d + e = 0
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 30
φh=90o
φa(t)t) ≠ const.
φr(t)t) = 90o - φ(t)t)
φb(t)t) ≠ const.
φc(t)t) = φ
a(t)t)
φd= 270o
φe= 180o
|r| = |AB| = r = const.
|h| = |OA| = h = const.
|a| = |OB| = a(t)t)
|c| = |OC| = c = const.
|b| = |CD| = b = const.
|d| = d(t)t)
|e| = e = const.
h + r = ac + b + d + e = 0
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 31
φh=90o
φa(t)t) ≠ const.
φr(t)t) = 90o - φ(t)t)
φb(t)t) ≠ const.
φc(t)t) = φ
a(t)t)
φd= 270o
φe= 180o
|r| = |AB| = r = const.
|h| = |OA| = h = const.
|a| = |OB| = a(t)t)
|c| = |OC| = c = const.
|b| = |CD| = b = const.
|d| = d(t)t)
|e| = e = const.
h + r = ac + b + d + e = 0
x : hcos ϕ h+r cosϕ r(t )=a(t)cosϕ a(t)y : hsin ϕ h+r sin ϕ r(t)=a(t)sin ϕ a(t)
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 32
φh=90o
φa(t)t) ≠ const.
φr(t)t) = 90o - φ(t)t)
φb(t)t) ≠ const.
φc(t)t) = φ
a(t)t)
φd= 270o
φe= 180o
|r| = |AB| = r = const.
|h| = |OA| = h = const.
|a| = |OB| = a(t)t)
|c| = |OC| = c = const.
|b| = |CD| = b = const.
|d| = d(t)t)
|e| = e = const.
h + r = ac + b + d + e = 0
x : hcos ϕ h+r cosϕ r(t )=a(t)cosϕ a(t)y : hsin ϕ h+r sin ϕ r(t)=a(t)sin ϕ a(t)
x : c cosϕ c(t)+bcosϕ b(t)+d(t)cosϕ d+e cosϕ e=0y : c sin ϕ c(t )+b sinϕ b (t)+d (t)sin ϕ d+e sin ϕ e=0
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 33
φh=90o
φa(t)t) ≠ const.
φr(t)t) = 90o - φ(t)t)
φb(t)t) ≠ const.
φc(t)t) = φ
a(t)t)
φd= 270o
φe= 180o
|r| = |AB| = r = const.
|h| = |OA| = h = const.
|a| = |OB| = a(t)t)
|c| = |OC| = c = const.
|b| = |CD| = b = const.
|d| = d(t)t)
|e| = e = const.
h + r = ac + b + d + e = 0
x : hcos ϕ h+r cosϕ r(t )=a(t)cosϕ a(t)y : hsin ϕ h+r sin ϕ r(t)=a(t)sin ϕ a(t)
x : c cosϕ c(t)+bcosϕ b(t)+d(t)cosϕ d+e cosϕ e=0y : c sin ϕ c(t )+b sinϕ b (t)+d (t)sin ϕ d+e sin ϕ e=0
x : hcos 90o+r cos(90o−ϕ(t ))=a(t)cosϕ a(t)
y : hsin 90o+r sin(90o−ϕ (t))=a(t)sin ϕ a(t)x : c cosϕ a(t)+bcosϕ b(t)+d (t)cos270
o+e cos180o=0
y : c sin ϕ a(t)+b sin ϕ b(t)+d (t)sin270o+e sin180o=0
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 34
φh=90o
φa(t)t) ≠ const.
φr(t)t) = 90o - φ(t)t)
φb(t)t) ≠ const.
φc(t)t) = φ
a(t)t)
φd= 270o
φe= 180o
|r| = |AB| = r = const.
|h| = |OA| = h = const.
|a| = |OB| = a(t)t)
|c| = |OC| = c = const.
|b| = |CD| = b = const.
|d| = d(t)t)
|e| = e = const.
h + r = ac + b + d + e = 0
x : hcos ϕ h+r cosϕ r(t )=a(t)cosϕ a(t)y : hsin ϕ h+r sin ϕ r(t)=a(t)sin ϕ a(t)
x : c cosϕ c(t)+bcosϕ b(t)+d(t)cosϕ d+e cosϕ e=0y : c sin ϕ c(t )+b sinϕ b (t)+d (t)sin ϕ d+e sin ϕ e=0
x : hcos 90o+r cos(90o−ϕ(t ))=a(t)cosϕ a(t)
y : hsin 90o+r sin(90o−ϕ (t))=a(t)sin ϕ a(t)x : c cosϕ a(t)+bcosϕ b(t)+d (t)cos270
o+e cos180o=0
y : c sin ϕ a(t)+b sin ϕ b(t)+d (t)sin270o+e sin180o=0
x : r sinϕ (t)=a(t)cosϕ a(t)y : h+r cosϕ (t )=a(t )sin ϕ a(t )x : c cosϕ a(t)+bcosϕ b(t)−e=0y : c sin ϕ a(t)+b sin ϕ b(t)−d (t)=0
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 35
x : r sinϕ (t)=a(t)cosϕ a(t)y : h+r cosϕ (t )=a(t )sin ϕ a(t )x : c cosϕ a(t)+bcosϕ b(t)−e=0y : c sin ϕ a(t)+b sin ϕ b(t)−d (t)=0
123
1
4
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 36
x : r sinϕ (t)=a(t)cosϕ a(t)y : h+r cosϕ (t )=a(t )sin ϕ a(t )x : c cosϕ a(t)+bcosϕ b(t)−e=0y : c sin ϕ a(t)+b sin ϕ b(t)−d (t)=0
a ( t )=√ r2 sin2( t )+(h+r cosϕ ( t ) )
2
ϕ a ( t )=atan( h+r cosϕ ( t )
r sin ϕ ( t ) )
ϕ b ( t )=arccose−c cosϕ a ( t )
b
d ( t )=c sin ϕ a ( t )+b √ 1−( e−c cosϕ a( t )
b )2
123
1
4
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 37
x : r sinϕ (t)=a(t)cosϕ a(t)y : h+r cosϕ (t )=a(t )sin ϕ a(t )x : c cosϕ a(t)+bcosϕ b(t)−e=0y : c sin ϕ a(t)+b sin ϕ b(t)−d (t)=0
a ( t )=√ r2 sin2( t )+(h+r cosϕ ( t ) )
2
ϕ a ( t )=atan( h+r cosϕ ( t )
r sin ϕ ( t ) )
ϕ b ( t )=arccose−c cosϕ a ( t )
b
vD ( t )=d ( t )
aD ( t )=d ( t )
123
1
4
d ( t )=c sin ϕ a ( t )+b √ 1−( e−c cosϕ a( t )
b )2
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 38
d ( t )=c sin ϕ a ( t )+b √ 1−( e−c cosϕ a( t )
b )2
r= 0,3 [m]
h= 0,4 [m]
c= 1 [m]
e= 0,4 [m]
b= 1 [m]
ω= 1 [rad/s]
a ( t )=√ r2 sin2( t )+(h+r cosϕ ( t ) )
2
ϕ ( t )
[m ]
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 39
ϕ a ( t )=atan( h+r cosϕ ( t )
r sin ϕ ( t ) )
ϕ b ( t )=arccose−c cosϕ a ( t )
b
ϕ ( t )
r= 0,3 [m]
h= 0,4 [m]
c= 1 [m]
e= 0,4 [m]
b= 1 [m]
ω= 1 [rad/s]
[rad ]
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 40
ϕ ( t )
r= 0,3 [m]
h= 0,4 [m]
c= 1 [m]
e= 0,4 [m]
b= 1 [m]
ω= 1 [rad/s]vD ( t )=d ( t )
ms
TM&AC, of driven element.Wint)er of driven element.2019/2020, of driven element.Sebast)ian of driven element.Korczak of driven element.PhD of driven element.Eng. 41
r= 0,3 [m]
h= 0,4 [m]
c= 1 [m]
e= 0,4 [m]
b= 1 [m]
ω= 1 [rad/s]
ϕ ( t )
aD ( t )=d ( t )
m
s2