Is this an allowed solution? - 2nd order harmonic oscillation

In summary, the conversation discusses the phase shift of input and output at resonance frequency, and the idea of using complex numbers to avoid piecemeal calculation and determine the phase shift. It also mentions that a vibrating system like a series LCR network may not naturally have an input and output, but one can be created by applying an input voltage across the R and taking an output voltage across L or C.
  • #1
APUGYael
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It is true that at resonance frequency the phase-shift between input and output is 90 degrees, so my mind would think that this is ok. But I am kind of unsure because of the whole dividing by zero part.

If this isn't allowed: is there any way to calculate/measure the damping coefficient with values for the damping ratio and resonance frequency? No, right?
 
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  • #2
What are you describing here - is it a vibrating system, like a series LCR network? In such a case , it does not naturally have an input and output.
However, if for example you apply an input voltage across the R and take an output voltage across L or C, then you see 90 deg phase shift.
 
  • #3
Yes, this is OK. The formula for ## tan \gamma ## is valid for all values of ## \omega ## except at resonance. However, at resonance, if you plug the value of ##\omega = \sqrt{\frac C J} ## into the original equation, the first and the third term cancel out and from the second term you get exactly 90 degrees phase shift.
There is a way to avoid this piecemeal calculation and that is using complex numbers. The force term is written as ## M exp^{i \omega \cdot t} ## and the response is ## \varphi = B \cdot exp^{i \omega \cdot t} ## with both, M and B being complex numbers. Differentiation is just multiplication by ## I\omega ## and the differential equation reduces to an algebraic equation. The phase shift is the argument of ## \frac {\varphi} M##
 

1. What is a 2nd order harmonic oscillation?

A 2nd order harmonic oscillation refers to a type of oscillatory motion where the restoring force is directly proportional to the displacement and acts in the opposite direction of the displacement. This results in a sinusoidal motion with a frequency that is twice the frequency of the original oscillation.

2. How is a 2nd order harmonic oscillation different from a 1st order harmonic oscillation?

A 1st order harmonic oscillation refers to a type of oscillatory motion where the restoring force is directly proportional to the displacement but acts in the opposite direction. This results in a sinusoidal motion with a frequency equal to the frequency of the original oscillation. In contrast, a 2nd order harmonic oscillation has a frequency that is twice the original frequency.

3. What factors determine whether a solution is allowed for a 2nd order harmonic oscillation?

The factors that determine whether a solution is allowed for a 2nd order harmonic oscillation include the initial conditions, the mass of the object, and the properties of the restoring force (such as the spring constant for a spring-mass system).

4. Can a 2nd order harmonic oscillation have multiple solutions?

Yes, a 2nd order harmonic oscillation can have multiple solutions depending on the initial conditions and the properties of the system. This is because the equation of motion for a 2nd order harmonic oscillation can have multiple roots, resulting in different solutions.

5. How is a 2nd order harmonic oscillation used in real-world applications?

2nd order harmonic oscillations are commonly observed in systems such as pendulums, springs, and electrical circuits. They are also used in various engineering applications, such as designing shock absorbers, tuning musical instruments, and stabilizing structures against vibrations.

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