MTH 235 Lab 05 April 16th, 2019
Nonlinear Pendulum
The equation for a pendulum as a first order differential system
Team Members: 1.
2.
3.
4.
Objectives
To have a better understanding of direction fields, phase portraits, and graph of solutions functions of the
equations describing a nonlinear pendulum.
Introduction
A pendulum consists of a small ball attached to the end of a rigid rod, the latter can swing from its other
end, as shown in the picture below. We assume that the ball at the end of the pendulum has mass m > 0
and the pendulum rod has length ` > 0. Also assume that the there is a friction force acting on the ball,
with damping coefficient d > 0. Recall that a friction force opposes the movement and is proportional to
the speed of the object.
The mass at the end of the pendulum will move on a circle of
radius ` centered at the oscillation point. Our main variable
is θ(t), the angle between the rod and the downward vertical,
positive in the counter clockwise direction. The displacement
of the pendulum mass along the the circle where it moves is `
s(t) = ` θ(t). Newton’s equation of motion is
θ
m
00 0
m (`θ) = −mg sin(θ) − d (`θ) .
The negative sign in the first term of the right-hand side above is because the component of the weight
tangent to the circle is in the direction opposite to the direction of increasing θ. The negative in the second
term above is because the friction force opposes the movement. We rewrite the equation as
g d
θ00 = − sin(θ) − θ0 .
` m
In this Lab we will focus on the case g/` = 1 and m = 1, that is
θ00 = − sin(θ) − d θ0 .
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MTH 235 Lab 05 April 16th, 2019
Critical Points and Linearization: No Friction
Consider the case of no friction, d = 0. Recall from lecture (Sect. 6.4) that the equilibrium points are the
same in the zero or nonzero friction case and are given by
(u, v) = (nπ, 0), where n is any integer.
If we evaluate the Jacobian matrix at the even critical points Ek = ((2k)π, 0), we find its eigenvalues are
λe± = ±i.
If we compute the Jacobian matrix at the odd critical points Ok = ((2k + 1)π, 0), we find that the
corresponding eigenvalues are λe± = ±1.
Question 1. Determine what the type of equilibrium at Ek and Ok , respectively. Do we have a sta-
ble/unstable spiral, stable/unstable node, saddle, center? Explain.
Graphical Analysis: No Friction Case
In the interactive graph below we show the direction field and solution curves for the first order reduction
of the nonlinear pendulum in the case g/` = 1 and m = 1, with 0 6 d < 2.
Interactive Graph Link.
Question 2. We consider the case of no friction. Use the interactive graph above with d = 0 and answer
the following:
(2a) Plot several solution curves and describe the two main kinds of qualitatively different solution curves.
(2b) What is the position of the pendulum in each type of critical point?
(2c) What is the physical interpretation of each type of solution curve in terms of motion of the pendulum.
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MTH 235 Lab 05 April 16th, 2019
Question 3. Use the interactive graph above with d = 0 and/or your physical intuition to answer the
following:
(3a) An initial condition which makes the pendulum make oscillations of up to π/4 around the lowest
position of the ball,
(3b) What is the initial velocity such that the pendulum starts at θ = 0 and reaches the unstable critical
point (π, 0) at infinite time? Can one obtain this solution experimentally?
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MTH 235 Lab 05 April 16th, 2019
Graphical Analysis: Small Friction Case
In the case of small friction, 0 < d < 2, we get d2 − 4 < 0, and one can find the following:
• For even critical points,
1 √
λe± = (−d ± i 4 − d2 ).
2
• For odd critical points,
1 √
λo± = (−d ± d2 + 4).
2
Question 4. Determine what the type of equilibrium at Ek = ((2k)π, 0) and Ok = ((2k + 1)π, 0), respec-
tively. Do we have a stable/unstable spiral, stable/unstable node, saddle, center? Explain.
Question 5. Use the interactive graph above with small friction, say d ' 0.2 and look at the solution
curves in this case for different initial conditions.
(5a) Is there an initial condition which makes the pendulum go round and round in full circles (infinitely
many times)? Explain based on the graph and based on the physics of the problem.
(5b) How do the solution curves change as the friction gets larger and larger? What does this mean in
terms of the motion of the pendulum?
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MTH 235 Lab 05 April 16th, 2019
Graphical Analysis: Large Friction Case
In the case of large friction, d > 2, we get d2 − 4 > 0, and one can find the following:
• For even critical points,
1 √
λe± = (−d ± d2 − 4).
2
• For odd critical points,
1 √
λo± = (−d ± d2 + 4).
2
Question 6. Determine what the type of equilibrium at Ek = ((2k)π, 0) and Ok = ((2k + 1)π, 0), respec-
tively. Do we have a stable/unstable spiral, stable/unstable node, saddle, center? Explain.
Question 7. For the case of large friction describe the behavior of the solution curves. What does this
mean in terms of the motion of the pendulum?
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