Trace-determinant plane: underdamped systems

In summary, to describe all underdamped systems with a period of 2 on the trace-determinant plane, we can use the equation T-4D=0, where T represents the period and D represents the determinant. Solving this equation will give us a parabola on the trace-determinant plane, and all points above this parabola will represent underdamped systems with a period of 2.
  • #1
clarineterr
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Homework Statement



Describe the set of points on the trace-determinant plane that describe all underdamped systems with period 2

Homework Equations



A harmonic oscillator can be represented by the equation m[tex]\frac{d^{2}y}{dt^{2}}[/tex] +b[tex]\frac{dy}{dt}[/tex]+ky=0 the Trace=-b/m and the Determinant=k/m.

For an underdamped oscillator, the period is 4m[tex]\pi[/tex]/[tex]\sqrt{4km-b^{2}}[/tex]
and T[tex]^{2}[/tex]-4D

The Attempt at a Solution



I have been trying to solve the equation 4m[tex]\pi[/tex]/[tex]\sqrt{4km-b^{2}}[/tex]=2 and then find T and D, but no luck really. I think the answer is all the points above a certain parabola.
 
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  • #2
I think T-4D=0 is the equation of the parabola, but I can't seem to prove it. Any help would be appreciated.
 

Related to Trace-determinant plane: underdamped systems

1. What is the trace-determinant plane in underdamped systems?

The trace-determinant plane is a representation of the behavior of underdamped systems in terms of their trace (the sum of their eigenvalues) and determinant (the product of their eigenvalues). It is a useful tool in analyzing the stability and dynamics of underdamped systems.

2. How is the trace-determinant plane used in underdamped systems?

The trace-determinant plane allows us to visualize the behavior of underdamped systems and determine their stability. By plotting the trace and determinant values of a system, we can determine the type of oscillatory behavior it will exhibit and whether it will approach a steady state or continue to oscillate.

3. What do the different regions in the trace-determinant plane represent?

In the trace-determinant plane, the regions are divided by a line called the boundary locus. The regions above the boundary locus represent stable systems, while the regions below it represent unstable systems. The boundary locus itself represents the critical damping value, where the system transitions from being underdamped to overdamped.

4. How does the location of eigenvalues affect the behavior of a system in the trace-determinant plane?

The location of eigenvalues in the trace-determinant plane determines the type of oscillatory behavior a system will exhibit. If the eigenvalues are in the stable region, the system will exhibit stable oscillations. If the eigenvalues are in the unstable region, the system will have unstable oscillations. If the eigenvalues are on the boundary locus, the system will exhibit critical damping.

5. Can the trace-determinant plane be used for other types of systems?

Yes, the trace-determinant plane can be used for a variety of linear systems, not just underdamped ones. It is a useful tool for analyzing the stability and dynamics of any system with two eigenvalues, including overdamped, critically damped, and complex conjugate eigenvalues.

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