SHM Math Explodes (Req: help finding Real parts)

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In summary, the conversation revolved around the topic of simple harmonic motion and solving ODEs related to it. The speaker was having trouble understanding the energy loss in a specific example and was seeking guidance. The conversation also touched on different resources and approaches to solving the problem. Ultimately, the expert confirms that the speaker is on the right track and provides some additional clarification.
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Homework Statement


I'm not attempting a specific problem, I'm just trying to find a correct way of doing the math behind simple harmonic motion ODEs...

The example problem I've given myself (based on the books I'm using) is a dashpot with constant [tex]c[/tex], spring with constant [tex]k[/tex] and a mass [tex]m[/tex].

Specifically, I'm trying to understand the energy lost by the dashpot, but I'm in the "Calculus" subforum for a reason:

Homework Equations



(variables [tex]/dt[/tex])
[tex]m\ddot x + c\dot x + kx = 0[/tex] - (Unforced, damped SHM).
which is eqivalent to:
[tex]\ddot x + \gamma\dot x + \omega_0^2 x = 0[/tex]

The solution is
[tex]x = \hat A e^{\alpha t}[/tex] and is differentiated & substituted back into get:

[tex]\alpha^2 + \gamma\alpha + \omega_0^2 = 0[/tex]

which is easy to solve for alpha:

[tex]\alpha = -\gamma / 2 \pm j\sqrt{\omega_0^2 - \gamma^2/4}[/tex]

The Attempt at a Solution


The problem I have is in substituting back in... at first it's not too bad, but I'm trying to get real values out... even with letting [tex]\omega_r = \sqrt{\omega_0^2 - \gamma^2/4}[/tex] it gets complicated quickly in my pen & paper scratches...

[tex]x = \hat A e^{-\gamma/2}(e^{j\omega_r t} + e^{-j\omega_r t})[/tex]

gets differentiated to find velocity (making it more complicated, but manageable), and then "de-Eulered" to find a real part. Am I doing something wrong? I don't see anything half as complicated in the books I'm using (primary book is "Fundamentals of Acoustics," by Kinsler and my additional resource is from Feynman's first volume on physics).

The books make it seem as though I'm doing something wrong, but I can't find how. Any help would be appreciated!
 
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  • #2
malweth said:
[tex]x = \hat A e^{-\gamma/2}(e^{j\omega_r t} + e^{-j\omega_r t})[/tex]

This is wrong. You've lost the decaying exponential part. Also, you only have one constant of integration. In a 2nd order ODE you must have 2 constants of integration in your general solution.

It should be:

[tex]x(t)=e^{-\gamma t/2}\left(Ae^{j\omega_rt}+Be^{-j\omega_rt}\right)[/tex]
 
  • #3
Thanks! I forgot about the other constant. The missing [tex]t[/tex] value in the exponent was a mistype.

Am I otherwise doing things correctly when trying to get back into real space (for Energy calculations)?
 
  • #4
It looks to me like you are.
 
  • #5
great! Thanks for the help.
 

1. What is SHM Math Explodes?

SHM Math Explodes is a mathematical concept that describes the motion of an object that moves back and forth repeatedly in a regular pattern.

2. Why is it important to find the real parts in SHM Math Explodes?

Finding the real parts in SHM Math Explodes is important because it helps us understand the physical meaning and behavior of the system being studied. It also helps us analyze and solve problems related to SHM.

3. What are the real parts in SHM Math Explodes?

The real parts in SHM Math Explodes refer to the physical quantities involved in the system, such as position, velocity, acceleration, and time. These real parts help us describe and analyze the motion of the system.

4. How can I find the real parts in SHM Math Explodes?

To find the real parts in SHM Math Explodes, you can use mathematical equations and formulas that describe the motion of the system. You can also plot graphs or use simulations to visualize the real parts.

5. What is the role of real parts in solving problems related to SHM Math Explodes?

The real parts play a crucial role in solving problems related to SHM Math Explodes. They help us understand the behavior and predict the motion of the system, as well as analyze the effects of different parameters on the system.

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