2.003SC
Recitation 7 Notes: Equations of Motion for Cart & Pendulum (Lagrange)
Cart and Pendulum - Problem Statement
A cart and pendulum, shown below, consists of a cart of mass, m1 , moving on a horizontal surface, acted
upon by a spring and damper with constants k and b , respectively. From the cart is suspended a pendulum
consisting of a uniform rod of length, l , and mass, m2 , pivoting about point A .
KINEMATICS: Write
• an expression for the linear velocity of point G, i.e. v G .
• an expression for the linear acceleration of point G, i.e. aG .
Cart and Pendulum - Kinematics
v G = ẋIˆ + 2l θ̇ĵ
aG = ẍIˆ + 2l θ¨ĵ + 2l θ̇ dtd ˆj
Recall that
dˆ
j
dt
= −θ̇î
aG = ẍIˆ − 2l θ̇2 î + 2l θ̈ĵ
In terms of the î and ĵ unit vectors,
l
l
aG = (ẍsinθ − θ̇2 )ˆi + (ẍcosθ + θ̈)ˆj
2
2
(1)
The figures below show the unit vectors, as well as point G’s velocity and acceleration components.
1
Cart and Pendulum - Free Body Diagrams
Draw the system’s free body diagrams.
Cart and Pendulum - Free Body Diagrams
2
Cart and Pendulum - Newton’s Laws
Summing forces on mass m2 in the î and ĵ -directions,
− F1 + m2 gcosθ = m2 aG
x
(2)
− F2 − m2 gsinθ = m2 aG
y
(3)
Equations (2) and (3) can be rewritten to isolate F1 and F2 , and the acceleration components from
equation (1) substituted.
l
F1 = m2 gcosθ − m2 axG = m2 gcosθ − m2 (ẍsinθ − θ̇2 )
2
l¨
F2 = −m2 gsinθ − m2 aG
y = −m2 gsinθ − m2 (ẍcosθ + θ)
2
Summing forces on mass m1 in the Iˆ -direction,
− kx − bẋ + F1 sinθ + F2 cosθ = m1 ẍ
(4)
Summing torques ABOUT POINT G on mass m2 in the +θ -direction,
F2
l
m2 l 2 ¨
=
θ
2
12
(5)
Substituting (2) and (3) into equations (4) and (5), we obtain,
l
l¨
−kx − bẋ + [m2 gcosθ − m1 (ẍsinθ − θ̇2 )]sinθ + [−m2 gsinθ − m1 (ẍcosθ + θ)]cosθ
= m1 ẍ
2
2
l¨ l
m2 l 2 ¨
=
θ
[−m2 gsinθ − m1 (ẍcosθ + θ)]
2 2
12
which (eventually) reduce to
(m1 + m2 )ẍ + bẋ + kx +
m2 l 2
m2 l ¨
θcosθ −
θ̇ sinθ = 0
2
2
(6)
and
(
l
m2 l2 m2 l2 ¨ m2 l
)θ +
¨
+
xcosθ
+ m2 g sinθ = 0
12
4
2
2
(7)
3
Summing torques about point A...
ΣA τ =
dA H
+ vA × P G
dt
l
P G = mv G = m(ẋIˆ + θ̇ˆj)
2
ΣA τ =
d G
[ H + r × P G] + vA × P G
dt
l
l
l
d m2 l 2
θ̇ + î × m2 (ẋIˆ + θ̇ĵ)] + ẋIˆ × m2 (ẋIˆ + θ̇ĵ)
Σ τ= [
dt 12
2
2
2
A
m2 l
m2 l 2
m2 l ˙
l
d m2 l 2
θ̇ +
xcosθ
˙
+
θ̇] +
x˙ θsinθ
−m2 g sinθ = [
2
dt 12
2
4
2
which (eventually) reduces to equation (7) above,
(
m2 l2 m2 l2 ¨ m2 l
l
+
)θ +
xcosθ
¨
+ m2 g sinθ = 0
12
4
2
2
4
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2.003SC / 1.053J Engineering Dynamics
Fall 2011
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