Prosiding KNEP V 2014 ISSN 2338-414X 434 pendulum back towards its equilibrium position. As the movement of the simple pendulum lifts it
through the equilibrium position, the restoring force converts its direction so that it is still directed towards the equilibrium position. A double pendulum is a system of two simple pendulum on a
common mounting which moves in anti-phase. In mathematics, in the area of dynamical system, a double pendulum is a pendulum with another pendulum attached to its end, and is a simple physical
system that exhibits rich dynamic behavior. The dynamics of a double pendulum system been studied since it has a close relationship with the motion of robot arm. The movement of a double pendulum is
adjusted by a set of paired ordinary differential equations. The value of angular position, angular velocity and acceleration was calculated by simulation in MATLABSimulink.
.
2. METODE
2.1. Double Pendulum Model
The double pendulum is a nonlinear system that exhibits chaos. This system contains two binary links that you connect with two revolute joints. Depending on its initial conditions, the double pendulum
can move in a quasi-periodic or chaotic fashion. Figure 2.1 shows the double pendulum as below,
Figure 2.1. Double Pendulum Model
2.2. Modelling Approach
Figure 2.2 shows the modeling approach of a double pendulum. To model the double pendulum, you represent each physical component
and constraint using a SimMechanics™ block. The double pendulum system contains three rigid bodies
—one pivot mount and two binary links— that connect in series through a pair of revolute joints. You represent the pivot mount and the binary links using the
custom library blocks that you created in previous examples. We can represent the two joints using two Revolute Joint blocks from the Joints library.
We can guide model assembly. By specifying joint state targets you can instruct SimMechanics to assemble a joint in the configuration we want. State targets that we can specify include position and
velocity, both angular and linear. At times, a state target may conflict with other state targets, or even with other kinematic constraints in the model. In these cases, you can prioritize the most important
state targets by assigning them a high priority level. During assembly, if two targets conflict with each other, SimMechanics assembles the high priority target first. To specify both state target values and
priority levels, we use the State Targets menu of the joint block dialog boxes.
Konferensi Nasional Engineering Perhotelan V, Universitas Udayana, 2014 435 Figure 2.2 Modelling approach of double pendulum in Simmechanics
2.3. Mathematic Modelling
Consider a double pendulum with masses m
1
= 0.15 kg and m
2
= 0.12 kg attached by rigid massless wires of lengths
� = 0.0 5 m and � = 0.0 m. Further, let the angles the two wires make with the vertical be denoted
� = 0. 9 rad and� = 0 radchanging with respect to time t, the system
exhibits 2 degrees of freedom as illustrated above. Finally, let gravity be given by g. Then the positions of the bobs are given by,
= � � = o
= � � � � = o o
The potential energy of this system is given by, = m m = m m o m o
The kinetic Energy by, =
= � �
� � � � � � � � o � � Energy is concerned, so the Lagrangian is
= =
m m m m o m m o m o
Prosiding KNEP V 2014 ISSN 2338-414X 436 So, for
� , = m
m o
= m m m o m n
= m m n m n
m m m o m m m n� = 0 For
� = m
m � � o
= m o m n
= m n n
so m o m n n� = 0
Dividing through l
2,
this simply to m o m n n� = 0
In the following analysis, the limbs are taken to be identical compound pendulums of length � and
mass , and the motion is restricted to two dimensions.
cos sin
sin sin
cos cos
2 1
2 2
2 1
1 2
1 2
1 2
2 2
2 2
2 1
2 2
1 2
1 1
2 1
2 1
m m
m l
g m
m L
m g
m L
m
o o
oo
sin sin
sin cos
cos sin
sin
2 1
2 2
1 2
2 1
2 1
2 1
2 2
2 1
2 1
2 2
2 1
2 1
1 2
1 2
m m
l g
g m
m l
m l
m m
o o
oo
3. RESULTS AND DISCUSSION 3.1. Design of Double Pendulum in MATLAB
