2) Consider the so-called Duffing oscillator which is an example of a damped and driven oscillator with a nonlinear restoring force. (A simple harmonic oscillator has a linear restoring force.) “Driven" just means there is an external force pushing/pulling on the end of the effective spring not connected to the oscillating mass. For example, think of your house as a kind of oscillator and the shaking ground due to an earthquake as the driving force. We'll use a slightly simplified version the relevant ODE: - - Ö – ß0(1 – 0²) = -αė + f coswt. = Here, a is the damping coefficient you would have if you had a plain old perfect Hookean spring (ẞ :0) while ẞ is a “stiffness” parameter describing the properties of the thing acting as the effective spring. We'll take beta = 0.5 for simplicity. The factor f charac- terizes the magnitude of the time varying forcing function and omega is its frequency. For specificity, use a = 0.2, f = 2 and w = 2πT. We leave out units since we do everything in the SI system of units. Make 2 subplots. The first will be displacement versus time. Label the axes sensibly and give the plot a sensible title. The second plot is a phase space plot. Label and title this subplot sensibly. When you make these plots a time span of (0, 100) may be useful, although other choices are fine. Make a sensible choice for the aspect ratio of the subplots.
2) Consider the so-called Duffing oscillator which is an example of a damped and driven oscillator with a nonlinear restoring force. (A simple harmonic oscillator has a linear restoring force.) “Driven" just means there is an external force pushing/pulling on the end of the effective spring not connected to the oscillating mass. For example, think of your house as a kind of oscillator and the shaking ground due to an earthquake as the driving force. We'll use a slightly simplified version the relevant ODE: - - Ö – ß0(1 – 0²) = -αė + f coswt. = Here, a is the damping coefficient you would have if you had a plain old perfect Hookean spring (ẞ :0) while ẞ is a “stiffness” parameter describing the properties of the thing acting as the effective spring. We'll take beta = 0.5 for simplicity. The factor f charac- terizes the magnitude of the time varying forcing function and omega is its frequency. For specificity, use a = 0.2, f = 2 and w = 2πT. We leave out units since we do everything in the SI system of units. Make 2 subplots. The first will be displacement versus time. Label the axes sensibly and give the plot a sensible title. The second plot is a phase space plot. Label and title this subplot sensibly. When you make these plots a time span of (0, 100) may be useful, although other choices are fine. Make a sensible choice for the aspect ratio of the subplots.
C++ for Engineers and Scientists
4th Edition
ISBN:9781133187844
Author:Bronson, Gary J.
Publisher:Bronson, Gary J.
Chapter7: Arrays
Section7.5: Case Studies
Problem 15E
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Please show the code for this assignment using python in a jupyter notebook. I am stuck
Transcribed Image Text:2) Consider the so-called Duffing oscillator which is an example of a damped and driven
oscillator with a nonlinear restoring force. (A simple harmonic oscillator has a linear
restoring force.) “Driven" just means there is an external force pushing/pulling on the
end of the effective spring not connected to the oscillating mass. For example, think of
your house as a kind of oscillator and the shaking ground due to an earthquake as the
driving force. We'll use a slightly simplified version the relevant ODE:
-
-
Ö – ß0(1 – 0²) = -αė + f coswt.
=
Here, a is the damping coefficient you would have if you had a plain old perfect Hookean
spring (ẞ :0) while ẞ is a “stiffness” parameter describing the properties of the thing
acting as the effective spring. We'll take beta = 0.5 for simplicity. The factor f charac-
terizes the magnitude of the time varying forcing function and omega is its frequency.
For specificity, use a = 0.2, f = 2 and w = 2πT. We leave out units since we do everything
in the SI system of units.
Make 2 subplots. The first will be displacement versus time. Label the axes sensibly
and give the plot a sensible title. The second plot is a phase space plot. Label and
title this subplot sensibly. When you make these plots a time span of (0, 100) may be
useful, although other choices are fine. Make a sensible choice for the aspect ratio of the
subplots.
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