In the second part, you will investigate what happens as the frequency of the E. In particular, you driving force w approaches the natural frequency wo = will consider the initial value problem u" + žu' + u = 5 cos(wt) (4) u(0) = 0, u'(0) = 0 for the values of w of 2, 1.5, and 1.1. You will need to solve the initial value problems for these three values of w (by hand or using Mathematica) and then graph the solutions using Mathematica. Use the graphs to comment on what is happening, particularly with the amplitude of any oscillations, particularly considering what happens as w approaches wo = 1. Compare your answer with the solution and graph of u" + u' +u = 5 cos t (5) u(0) = 0, u'(0) = 0
In the second part, you will investigate what happens as the frequency of the E. In particular, you driving force w approaches the natural frequency wo = will consider the initial value problem u" + žu' + u = 5 cos(wt) (4) u(0) = 0, u'(0) = 0 for the values of w of 2, 1.5, and 1.1. You will need to solve the initial value problems for these three values of w (by hand or using Mathematica) and then graph the solutions using Mathematica. Use the graphs to comment on what is happening, particularly with the amplitude of any oscillations, particularly considering what happens as w approaches wo = 1. Compare your answer with the solution and graph of u" + u' +u = 5 cos t (5) u(0) = 0, u'(0) = 0
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Considering the problem, do not worry about the graphs. Just need help with the math, please.
Transcribed Image Text:Varying Driving Frequency
In the second part, you will investigate what happens as the frequency of the
E. In particular, you
driving force w approaches the natural frequency wo =
will consider the initial value problem
Su" + }u' +u = 5 cos(wt)
(4)
lu(0) = 0, u'(0) = 0
for the values of w of 2, 1.5, and 1.1. You will need to solve the initial value
problems for these three values of w (by hand or using Mathematica) and then
graph the solutions using Mathematica. Use the graphs to comment on what
is happening, particularly with the amplitude of any oscillations, particularly
considering what happens as w approaches wo = 1. Compare your answer with
the solution and graph of
u" + u' +u = 5 cos t
(5)
| u(0) = 0, u'(0) = 0
Transcribed Image Text:1
The Problem
In class, we discussed the undamped, forced oscillator. In this project, you will
consider the damped, forced oscillator problem
(1) mu" + yu' + ku = Fo cos(wt).
In this project, we will consider changing y and changing w.
2
Your Task
Your project is divided into two parts.
2.1
Varying Dampening
The first part is to consider the initial value problem
(u" + yu' +u = 5 cos(2t)
(2)
u(0) = 0, u'(0) = 0
for the following values of y: 2, 1, }, and . Note that we are taking the spring
constant, mass of the block, and amplitude of the driving force to be one. You
will need to solve these three problems either by hand or using Mathematica
1
(your choice) and then use Mathematica to plot the solutions. In your document,
you will need to comment on what you observe is happening to the solution as
y becomes smaller. It would be helpful to compare (hint, hint) the solution
graphs with the graph of the solution to the problem without dampening
u" + u = 5 cos(2t)
(3)
| u(0) = 0, u'(0) = 0.
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