How to Determine Damping Ratio and Pole Zeros from a Differential Equation?

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To determine the damping ratio and pole zeros from the given differential equation, a forward Laplace transform is applied, leading to the transfer function Y(s) = 63/(s^2 - 57s + 10). The poles are calculated using the quadratic formula, resulting in s = 28.5 +/- 56.648, indicating potential issues with expected complex roots. The discussion emphasizes that the input magnitude does not affect the pole and zero locations or the damping ratio, as these are inherent to the system's transfer function H(s). The analysis concludes that the stability and behavior of the system can be assessed through the pole-zero diagram derived from H(s).
Steve Collins
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The following diff. equation describes the functionality of a system with respect to time. However, it is not known how the system will behave when stimulated. Apply a forward Laplace transform to determine damping ratio and pole zeros. Plot a pole zero diagram and comment on stability.

d2y/dt2 + 6dy/dt + 10y = 63dy/dt + 63xPutting like terms on either side of the equation:

d2y/dt2 + 6dy/dt - 63dy/dt + 10y = 63x

= d2y/dt2 - 57dy/dt + 10y = 63x

Laplace transform:

s2Y(s) - 57sY(s) + 10Y(s) = 63 ... (63 because unit impulse used to stimulate?)

Simplify for Y(s):

Y(s).(s2 - 57s + 10) = 63

Solve for Y(s):

Y(s) = 63/(s2 - 57s + 10)

Using quadratic formula to find poles:

s= (57 +/- √572 - 4 x 1 x 10)/(2 x 1)

= (57 +/- 56.648)/2 = 28.5 +/- 56.648 (I was expecting a complex number!)

I think that I am nearly there, but I suspect that I have gone wrong.
 
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Steve Collins said:
The following diff. equation describes the functionality of a system with respect to time. However, it is not known how the system will behave when stimulated. Apply a forward Laplace transform to determine damping ratio and pole zeros. Plot a pole zero diagram and comment on stability.

d2y/dt2 + 6dy/dt + 10y = 63dy/dt + 63x


Putting like terms on either side of the equation:

d2y/dt2 + 6dy/dt - 63dy/dt + 10y = 63x

= d2y/dt2 - 57dy/dt + 10y = 63x

Laplace transform:

s2Y(s) - 57sY(s) + 10Y(s) = 63 ... (63 because unit impulse used to stimulate?)

Factor out Y(s)F(s) = X(s) from the above transformed equation with X(s) = 1 (unit impulse input).

Then Y(s)/X(s) = 1/F(s) and let H(s) = 1/F(s) so that now Y(s) = X(s)H(s). H(s) is now the system transfer function. Given any X(s) you can now compute Y(s) and consequently y(t).

Note that the magnitude of the input impulse has nothing to do with pole/zero location, nor damping ratio. You only use it if you want to compute Y(s) and y(t).

Note also that with an impulse input, all the poles & zeros of the output Y(s) are due to the system H(s) only.
 

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