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We can calculate the period and half-amplitude time for phugoid and short period using equations of motion. We will get an exact answer.

Then why is there a need for doing approximations to equations of motion for calculating period and half-amplitude time for both motions?

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    $\begingroup$ Hello Pavankumar Koratikere, welcome to aviation.stackexchange. I've rephased the title of your question into an actual question, which is the standard here. $\endgroup$ – DeltaLima Nov 6 '19 at 14:46
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By approximations, I believe you mean the following (found in any major stability & control textbook and this MIT course note), where the $_0$ notation stands for undisturbed trim condition:

  • Short-Period damping:

$$\zeta_{sp}\approx-\frac{M_q}{2}\sqrt{-\frac{1}{V_0M_wI_{yy}}}$$

  • Short-Period frequency:

$$\omega_{sp}\approx\sqrt{-\frac{V_0M_w}{I_{yy}}}$$

  • Phugoid damping:

$$\zeta_{p}\approx\frac{1}{\sqrt{2}}\frac{C_{D_0}}{C_{L_0}}$$

  • Phugoid frequency:

$$\omega_{p}\approx\sqrt{2}\frac{g}{V_0}$$

They are useful because:

  1. They are computationally easy, and offer first approximation during early conceptual design and rapid iterations. However, in the 21st century, that's not saying too much.
  2. They offer important design insight to engineers. It's not enough to calculate the modes; more important is to understand how to change the aircraft design such that the modes are adequate. With the full linearized matrix, it's hard to appreciate how derivatives influence these results (certainly for me). Now when you look at these simpler equations, the effects are almost intuitive! For this reason, these approximations (especially the lateral/directional approximations) will never go out of fashion.

On another note, since aerodynamics, especially transonic aerodynamics, propulsion and the equations of motions are inherently nonlinear, the linearized matrix (and the concept of eigenmodes) is itself an approximation.

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    $\begingroup$ That last paragraph is indeed very important. All mathematical models are approximations. The only way to get to the exact period and half-amplitude time is to measure it in flight. $\endgroup$ – DeltaLima Nov 6 '19 at 14:48

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