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Spring-Cart-Pendulum System

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animation
slow
normal
fast
plot parameters
x[t] - θ[t]
x'[t] - θ'[t]
pendulum parameters
length of rod L
1.5
mass of bob m
1
starting angle
θ
0
85°
cart parameters
mass of cart c
1
starting position
x
0
1
spring constant k
10
This Demonstration models the dynamical behavior of a pendulum connected to a cart that is, in turn, attached to a spring. The path of the pendulum bob and the position and speed of the cart and the pendulum can be traced as you vary the parameters of the system. The spring-cart-pendulum system has two degrees of freedom: the horizontal position of the cart and pendulum
x(t)
and the angle between the pendulum's rod and the vertical
θ(t)
. In the absence of friction, we have the following equations of motion (
g
is acceleration due to gravity):
Lm
2
′
θ
(t)
sin(θ(t))-
′′
θ
(t)cos(θ(t))(c+m)
′′
x
(t)+kx(t)
,
gsin(θ(t))+L
′′
θ
(t)+cos(θ(t))
′′
x
(t)0
.
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