8. Suppose we take l = 9.8 m (so g/l = 1), m = 1, and b large, say b = 4. For the damped pendulum system above and with this choice of parameter values, do the following. (a) Find the cigenvalues and eigenvectors of the linearized system at the equilib- rium point (0, 0). (b) Find the eigenvalues and eigenvectors of the linearized system at the equilib- rium point (7, 0). (c) Sketch the phase portrait near the equilibrium points. (d) Sketch the entire phase portrait. [Hint: Begin by sketching the level sets of H as in the text.]
8. Suppose we take l = 9.8 m (so g/l = 1), m = 1, and b large, say b = 4. For the damped pendulum system above and with this choice of parameter values, do the following. (a) Find the cigenvalues and eigenvectors of the linearized system at the equilib- rium point (0, 0). (b) Find the eigenvalues and eigenvectors of the linearized system at the equilib- rium point (7, 0). (c) Sketch the phase portrait near the equilibrium points. (d) Sketch the entire phase portrait. [Hint: Begin by sketching the level sets of H as in the text.]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Solve all the subparts with clear handwriting
Transcribed Image Text:In Exercises 4-11, we consider the damped pendulum system
de
dt
ap
dt
b
sine
-v,
m
where b is the damping coefficient, m is the mass of the bob, I is the length of the arm,
and g is the acceleration of gravity (g 9.8 m/s?).
![8. Suppose we take / = 9.8 m (so g/I = 1), m 1, and b large, say b = 4. For the
damped pendulum system above and with this choice of parameter values, do the
following.
(a) Find the cigenvalues and eigenvectors of the linearized system at the equilib-
rium point (0, 0).
(b) Find the eigenvalues and eigenvectors of the linearized system at the equilib-
rium point (7, 0).
(c) Sketch the phase portrait near the equilibrium points.
(d) Sketch the entire phase portrait. [Hint: Begin by sketching the level sets of H
as in the text.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5bd537f3-9c5e-43d2-a2ef-ea54935c61b7%2F8f080b82-c2b3-4187-9b8a-c3b188c7eff9%2F1j1yqw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:8. Suppose we take / = 9.8 m (so g/I = 1), m 1, and b large, say b = 4. For the
damped pendulum system above and with this choice of parameter values, do the
following.
(a) Find the cigenvalues and eigenvectors of the linearized system at the equilib-
rium point (0, 0).
(b) Find the eigenvalues and eigenvectors of the linearized system at the equilib-
rium point (7, 0).
(c) Sketch the phase portrait near the equilibrium points.
(d) Sketch the entire phase portrait. [Hint: Begin by sketching the level sets of H
as in the text.]
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