Finding the amplitude an oscillator: Driven harmonic oscillator problem

In summary, the conversation is about a car moving along a hill with a small slope, connected to an upper mass by a spring and a damper. The upper mass's displacement is described by a differential equation. The profile of the road surface is given and the task is to find the amplitude of the oscillations of the upper mass. The conversation involves equations and the website of the professor's hint is provided. The solution involves calculating the magnitude of a variable c.
  • #1
Benzoate
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Homework Statement



A car is moving along a hill at constant speed on an undulating road with profile h(x) where h'(x) is small. The car is represented by a chassis which keeps contact with the road , connected to an upper mass m by a spring and a damper. At time t, the upper mas has displacement y(t) satisfies a differential equation of the form

y(double dot) + 2Ky(single dot)+[tex]\Omega[/tex]2= 2Kch'(ct) + [tex]\Omega[/tex]2h(ct)

where K and [tex]\Omega[/tex] are positve constants.

Suppose that the profile of the road surface is given by h(x) = h0cos(px/c), where h0 and p are positive constants. Find the amplitude a of the driven oscillations of the upper mass.

I will post the website that contains my professor's hint to this problem and since the hint is in pdf form, I am unable to paste it

http://courses.ncsu.edu/py411/lec/001/ [Broken]

Go to homework tab
Then go to assignment 7
then go to 5.11 once you've clicked on assignment 7



Homework Equations






The Attempt at a Solution



y(double dot)+2K[tex]\varsigma[/tex]'+[tex]\Omega[/tex]2[tex]\varsigma[/tex]=0

y=h(ct)+[tex]\varsigma[/tex]==> [tex]\varsigma[/tex]=y-h(ct)

[tex]\varsigma[/tex](single dot)=y(single dot)-h'c(ct)
y=ceipt
y(single dot)=cipeipt
y(double dot)=-c^2eipt
since h(x)=h(ct) and h(x) = h0cos(px/c),then h(x)= h0cos(px/c)= h0cos(pt)

h(x)= h0cos(pt)
h'(x)=-p h0sin(pt)

could I say h(x)= h0cos(pt)=h0e^ipt?

then
h(x)=h0e^ipt
h'(x)=iph0e^ipt

therefore, [tex]\varsigma[/tex]=y-h(ct) becomes [tex]\varsigma[/tex]=y-h(pt)=> [tex]\varsigma[/tex](single dot)=y(single dot)-h'p(pt)

plugging all of my variables into the equation y(double dot) + 2Ky(single dot)+[tex]\Omega[/tex]2= 2Kch'(ct) + [tex]\Omega[/tex]2h(ct)

I find c to be :

c=h0(2kp^2+[tex]\Omega[/tex]2)/([tex]\Omega[/tex]2+2ki-2kh0p)

I do realize in order to get the amplitude I have to calculate the magnitude of c: I think I calculated my magnitude incorrectly :

According to my textbook , here is the actually amplitude

a= (([tex]\Omega[/tex]4+4K^2p^2)/(([tex]\Omega[/tex]2-p^2)2+4K^2p^2))1/2
 
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anybody have any trouble reading my solution?
 

What is an oscillator?

An oscillator is a physical system that exhibits periodic motion around an equilibrium point.

What is a driven harmonic oscillator?

A driven harmonic oscillator is an oscillator that is subjected to an external force that causes it to oscillate at a particular frequency.

How do you find the amplitude of a driven harmonic oscillator?

The amplitude of a driven harmonic oscillator can be found by solving the differential equation that describes its motion and using the initial conditions to determine the amplitude at a particular time.

What factors affect the amplitude of a driven harmonic oscillator?

The amplitude of a driven harmonic oscillator is affected by the amplitude and frequency of the external force, as well as the natural frequency and damping coefficient of the oscillator itself.

What is the role of resonance in a driven harmonic oscillator?

Resonance occurs when the driving frequency of an external force matches the natural frequency of the oscillator, resulting in a large amplitude of oscillation. This can lead to unstable behavior and potential damage to the oscillator system.

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